Theory of functions of a complex variable

•

ill

THEOBY OF FUNCTIONS

OF A

COMPLEX VARIABLE.

Itonlion: C. J. CLAY AND SONS,

CAMBEIDGE UNIVEESITY PEESS WAEEHOUSE,

AVE MAEIA LANE.

CAMBEIDGE : DEIGHTON, BELL, AKD CO.

LEIPZIG : F. A. BROCKHAUS.
NEW YORK: MACMILLAN AND CO.

THEOEY OF FUNCTIONS

OF A

COMPLEX VARIABLE

BY

A. R FORSYTE, So.D., F.RS.,

FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

CAMBEIDGE:
AT THE UNIVERSITY PRESS.

1893

All rights reserved. .

Mtth. U. 01.

PRINTED BY C. J. CLAY, M.A. AND SONS,
AT THE UNIVERSITY PRESS.

PEEFACE.

AMONG the many advances in the progress of mathematical
XlL science during the last forty years, not the least remarkable
are those in the theory of functions. The contributions that are
still being made to it testify to its vitality : all the evidence points
to the continuance of its growth. And, indeed, this need cause no
surprise. Few subjects can boast such varied processes, based
upon methods so distinct from one another as are those originated
by Cauchy, by Weierstrass, and by Biemann. Each of these
methods is sufficient in itself to provide a complete development ;
combined, they exhibit an unusual wealth of ideas and furnish
unsurpassed resources in attacking new problems.

It is difficult to keep pace with the rapid growth of the
literature which is due to the activity of mathematicians,
especially of continental mathematicians : and there is, in con
sequence, sufficient reason for considering that some marshalling
of the main results is at least desirable and is, perhaps, necessary.
Not that there is any dearth of treatises in French and in
German : but, for the most part, they either expound the pro
cesses based upon some single method or they deal with the
discussion of some particular branch of the theory.

814033

PREFACE

The present treatise is an attempt to give a consecutive
account of what may fairly be deemed the principal branches of
the whole subject. It may be that the next few years will see
additions as important as those of the last few years : this account
would then be insufficient for its purpose, notwithstanding the
breadth of range over which it may seem at present to extend.
My hope is that the book, so far as it goes, may assist mathe
maticians, by lessening the labour of acquiring a proper knowledge
of the subject, and by indicating the main lines, on which recent
progress has been achieved.

No apology is offered for the size of the book. Indeed, if
there were to be an apology, it would rather be on the ground
of the too brief treatment of some portions and the omissions
of others. The detail in the exposition of the elements of several
important branches has prevented a completeness of treatment
of those branches : but this fulness of initial explanations is
deliberate, my opinion being that students will thereby become
better qualified to read the great classical memoirs, by the study
of which effective progress can best be made. And limitations of
space have compelled me to exclude some branches which other
wise would have found a place. Thus the theory of functions of
a real variable is left undiscussed : happily, the treatises of Dini,
Stolz, Tannery and Chrystal are sufficient to supply the omission.
Again, the theory of functions of more than one complex variable
receives only a passing mention ; but in this case, as in most
cases, where the consideration is brief, references are given
which will enable the student to follow the development to
such extent as he may desire. Limitation in one other direction
has been imposed : the treatise aims at dealing with the general
theory of functions and it does not profess to deal with special
classes of functions. I have not hesitated to use examples of
special classes : but they are used merely as illustrations of the
general theory, and references are given to other treatises for
the detailed exposition of their properties.

PREFACE Vll

The general method which is adopted is not limited so that
it may conform to any single one of the three principal inde
pendent methods, due to Cauchy, to Weierstrass and to Biemann
respectively : where it has been convenient to do so, I have
combined ideas and processes derived from different methods.

The book may be considered as composed of five parts.

The first part, consisting of Chapters I — VII, contains the
theory of uniform functions : the discussion is based upon power-
series, initially connected with Cauchy's theorems in integration,
and the properties established are chiefly those which are con
tained in the memoirs of Weierstrass and Mittag-Leffler.

The second part, consisting of Chapters VIII — XIII, contains
the theory of multiform functions, and of uniform periodic
functions which are derived through the inversion of integrals
of algebraic functions. The method adopted in this part is
Cauchy's, as used by Briot and Bouquet in their three memoirs
and in their treatise on elliptic functions : it is the method that
has been followed by Hermite and others to obtain the properties
of various kinds of periodic functions. A chapter has been
devoted to the proof of Weierstrass's results relating to functions
that possess an addition-theorem.

The third part, consisting of Chapters XIV — XVIII, contains
the development of the theory of functions according to the
method initiated by Biemann in his memoirs. The proof which
is given of the existence-theorem is substantially due to Schwarz ;
in the rest of this part of the book, I have derived great assist
ance from Neumann's treatise on Abelian functions, from Fricke's
treatise on Klein's theory of modular functions, and from many
memoirs by Klein.

The fourth part, consisting of Chapters XIX and XX, treats
of conformal representation. The fundamental theorem, as to the
possibility of the conformal representation of surfaces upon one
another, is derived from the existence-theorem : it is a curious fact
that the actual solution, which has been proved to exist in general,
F. b

Vlll PREFACE

has been obtained only for cases in which there is distinct
limitation.

The fifth part, consisting of Chapters XXI and XXII, contains
an introduction to the theory of Fuchsian or automorphic functions,
based upon the researches of Poincare and Klein : the discussion is
restricted to the elements of this newly-developed theory.

The arrangement of the subject-matter, as indicated in this
abstract of the contents, has been adopted as being the most
convenient for the continuous exposition of the theory. But the
arrangement does not provide an order best adapted to one who is
reading the subject for the first time. I have therefore ventured
to prefix to the Table of Contents a selection of Chapters that
will probably form a more suitable introduction to the subject for
such a reader ; the remaining Chapters can then be taken in an
order determined by the branch of the subject which he wishes
to follow out.

In the course of the preparation of this book, I have consulted
many treatises and memoirs. References to them, both general
and particular, are freely made : without making precise reserva
tions as to independent contributions of my own, I wish in this
place to make a comprehensive acknowledgement of my obligations
to such works. A number of examples occur in the book : most of
them are extracted from memoirs, which do not lie close to the
direct line of development of the general theory but contain
results that provide interesting special illustrations. My inten
tion has been to give the author's name in every case where a
result has been extracted from a memoir : any omission to do so
is due to inadvertence.

Substantial as has been the aid provided by the treatises and
memoirs to which reference has just been made, the completion of
the book in the correction of the proof-sheets has been rendered
easier to me by the unstinted and untiring help rendered by
two friends. To Mr William Burnside, M.A., formerly Fellow of

PREFACE

Pembroke College, Cambridge, and now Professor of Mathematics
at the Royal Naval College, Greenwich, I am under a deep debt
of gratitude : he has used his great knowledge of the subject in
the most generous manner, making suggestions and criticisms that
have enabled me to correct errors and to improve the book in
many respects. Mr H. M. Taylor, M. A., Fellow of Trinity College,
Cambridge, has read the proofs with great care : the kind assist
ance that he has given me in this way has proved of substantial
service and usefulness in correcting the sheets. I desire to
recognise most gratefully my sense of the value of the work which
these gentlemen have done.

It is but just on my part to state that the willing and active
co-operation of the Staff of the University Press during the pro
gress of printing has done much to lighten my labour.

It is, perhaps, too ambitious to hope that, on ground which
is relatively new to English mathematics, there will be freedom
from error or obscurity and that the mode of presentation in this
treatise will command general approbation. In any case, my aim
has been to produce a book that will assist mathematicians in
acquiring a knowledge of the theory of functions : in proportion
as it may prove of real service to them, will be my reward.

A. R. FORSYTE.

TRINITY COLLEGE, CAMBRIDGE.
25 February, 1893.

CONTENTS.

The following course is recommended, in the order specified, to those who are
reading the subject for the first time : The theory of uniform functions, Chapters
I— V ; Conformal representation, Chapter XIX ; Multiform functions and uniform
periodic functions, Chapters VIII— XI ; Riemanris surfaces, and Riemann's theory
of algebraic functions and their integrals, Chapters XIV— XVI, XVIII.

CHAPTER I.

GENERAL INTRODUCTION.

§§

PAGE

1—3. The complex variable and the representation of its variation by points

in a plane ,

4. Neumann's representation by points on a sphere ... 4

5. Properties of functions assumed known ... Q
6, 7. The idea of complex functionality adopted, with the conditions neces
sary and sufficient to ensure functional dependence ... 6

8. Riemann's definition of functionality ... g

9. A functional relation between two complex variables establishes the

geometrical property of conformal representation of their planes . 10

10, 11. Relations between the real and the imaginary parts of a function of z 11
12, 13. Definitions and illustrations of the terms monogenic, uniform, multiform,

branch, branch-point, holomorphic, zero, pole, meromorphic . . . 14

CHAPTER II.

INTEGRATION OF UNIFORM FUNCTIONS.

14, 15. Definition of an integral with complex variables ; inferences . . . ' 18
16. Proof of the lemma I I (^ - £ \ dxdy=\(pdx -\-qdy), under assigned

I \ fll'1 (it I I J •* J. •/ f ' O

conditions 21

CONTENTS

§§ PAGE

17, 18. The integral \f(z)dz round any simple curve is zero, when f(z) is

Cz

holomorphic within the curve; and I /(*)<& is a holomorphic

J a

function when the path of integration lies within the curve . . 23
19. The path of integration of a holomorphic function can be deformed

without changing the value of the integral ..... 26

20—22. The integral =— . I '— '- dz, round a curve enclosing a, is /(a) when
27rt J z — a

f(z) is a holomorphic function within the curve; and the integral

J_ [ /(*) dz is — ,—^. Superior limit for the modulus of
27rt J(z-a)n + 1 n\ dan

the nth derivative of /(a) in terms of the modulus of /(a) . . 27

23. The path of integration of a meromorphic function cannot be deformed

across a pole without changing the value of the integral. . . 34

24. The integral of any function (i) round a very small circle, (ii) round a

very large circle, (iii) round a circle which encloses all its infinities

and all its branch-points ......... 35

25. Special examples ............ ••

CHAPTER III.

EXPANSIONS OF FUNCTIONS IN SERIES OF POWERS.

26, 27. Cauchy's expansion of a function in positive powers of z - a ; with re
marks and inferences 43

28—30. Laurent's expansion of a function in positive and negative powers of

z - a ; with corollary 47

31. Application of Cauchy's expansion to the derivatives of a function . 51

32, 33. Definition of an ordinary point of a function, of the domain of an
ordinary point, of an accidental singularity, and of an essential
singularity .......••••• 52

34, 35. Continuation of a function by means of elements over its region of

continuity 54

36. Schwarz's theorem on symmetric continuation across the axis of real

quantities 57

CHAPTER IV.

UNIFORM FUNCTIONS, PARTICULARLY THOSE WITHOUT ESSENTIAL
SINGULARITIES.

37. A function, constant over a continuous series of points, is constant

everywhere in its region of continuity 59

38, 39. The multiplicity of a zero, which is an ordinary point, is finite; and

a multiple zero of a function is a zero of its first derivative . . 61

CONTENTS Xlll

§§ PAGE

40. A function, that is not a constant, must have infinite values . . 63

41, 42. Form of a function near an accidental singularity 64

43, 44. Poles of a function are poles of its derivatives ..... 66
45, 46. A function, which has infinity for its only pole and has no essential

singularity, is an algebraical polynomial ...... 69

47. Integral algebraical and integral transcendental functions ... 70

48. A function, all the singularities of which are accidental, is an algebraical

meromorphic function .......... 71

CHAPTER V.

TRANSCENDENTAL INTEGRAL UNIFORM FUNCTIONS.

49, 50. Construction of a transcendental integral function with assigned zeros
a1? a2, a3, ..., when an integer s can be found such that 2|an|~8

is a converging series 74

51. Weierstrass's construction of a function with any assigned zeros . . 77
52, 53. The most general form of function with assigned zeros and having

its single essential singularity at 0=00 . . . . . . 80

54. Functions with the singly-infinite system of zeros given by ;Z = TO<B, for

integral values of m 82

55 — 57. Weierstrass's o--function with the doubly-infinite system of zeros given

by z=ma> + m'a>, for integral values of TO and of TO' . . . . 84
58. A function cannot exist with a triply-infinite arithmetical system of zeros 88

59, 60. Class (genre) of a function 89

61. Laguerre's criterion of the class of a function 91

CHAPTER VI.

FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES.

62. Indefiniteness of value of a function at an essential singularity . . 94

63. A function is of the form O { — =- ) + P (z — 6) in the vicinity of an essen-

\»— o/

tial singularity at b, a point in the finite part of the plane . . 96
64, 65. Expression of a function with n essential singularities as a sum of n

functions, each with only one essential singularity .... 99
66, 67. Product-expression of a function with n essential singularities and no

zeros or accidental singularities 101

68 — 71. Product-expression of a function with n essential singularities and with

assigned zeros and assigned accidental singularities ; with a note

on the region of continuity of such a function . . . .104

xiv CONTENTS

CHAPTER VII.

FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES,
AND EXPANSION IN SERIES OF FUNCTIONS.

§§ 1>AGE

72. Mittag-Leffler's theorem on functions with unlimited essential singu

larities, distributed over the whole plane 112

73. Construction of subsidiary functions, to be terms of an infinite sum . 113
74_76. Weierstrass's proof of Mittag-Leffler's theorem, with the generalisation

of the form of the theorem 114

77, 78. Mittag-Leffler's theorem on functions with unlimited essential singu
larities, distributed over a finite circle 117

79. Expression of a given function in Mittag-Leffler's form .... 123

80. General remarks on infinite series, whether of powers or of functions . 126

81. A series of powers, in a region of continuity, represents one and only

one function ; it cannot be continued beyond a natural limit . . 128

82. Also a series of functions : but its region of continuity may consist of

distinct parts 129

83. A series of functions does not necessarily possess a derivative at points

on the boundary of any one of the distinct portions of its region

of continuity ........••• 133

84. A series of functions may represent different functions in distinct parts

of its region of continuity ; Tannery's series 136

85. Construction of a function which represents different assigned functions

in distinct assigned parts of the plane . . . . . .138

86. Functions with a line of essential singularity 139

87. Functions with an area of essential singularity or lacunary spaces . 141

88. Arrangement of singularities of functions into classes and species . . 146

CHAPTER VIII.

MULTIFORM FUNCTIONS.

89. Branch-points and branches of functions 149

90. Branches obtained by continuation: path of variation of independent

variable between two points can be deformed without affecting a
branch of a function if it be not made to cross a branch-point . 150

91. If the path be deformed across a branch-point which affects the branch,

then the branch is changed I55

92. The interchange of branches for circuits- round a branch-point is cyclical 156

93. Analytical form of a function near a branch-point 157

94. Branch-points of a function defined by an algebraical equation in their

relation to the branches : definition of algebraic function . . 161
95, 96. Infinities of an algebraic function 163

CONTENTS xv

PAGE

97. Determination of the branch-points of an algebraic function, and of the

cyclical systems of the branches of the function ... 168

98. Special case, when the branch-points are simple : their number . . 174

99. A function, with n branches and a limited number of branch-points and

singularities, is a root of an algebraical equation of degree n. . 175

CHAPTER IX.

PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS
IN GENERAL.

100. Conditions under which the path of variation of the integral of a

multiform function can be deformed without changing the value

of the integral ....... J§Q

101. Integral of a multiform function round a small curve enclosing a

branch-point ....... 183

102. Indefinite integrals of uniform functions with accidental singularities ;

fdz f dz

j i ' 2 ..... • ..... 184

103. Hermite's method of obtaining the multiplicity in value of an integral;

sections in the plane, made to avoid the multiplicity . . .185

104. Examples of indefinite integrals of multiform functions ; \wdz round

any loop, the general value of J(l - 22) ~ * dz, of J{1 - z2) (1 - k^}} ~ * dz,

and of 5{(z-el)(z-e2)(g-es)}-*dz ....... 189

105. Graphical representation of simply-periodic and of doubly-periodic

functions ....... 198

106. The ratio of the periods of a uniform doubly-periodic function is not

real ............. 200

107, 108. Triply-periodic uniform functions of a single variable do not exist . 202

109. Construction of a fundamental parallelogram for a uniform doubly-

periodic function ....... 205

110. An integral, with more periods than two, can be made to assume any

value by a modification of the path of integration between the
limits ........ 208

CHAPTER X.

SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS.

2rrzi

111. Simply-periodic functions, and the transformation Z=e w . ;' • . 211

112. Fourier's series and simply-periodic functions 213

113, 114. Properties of simply-periodic functions without essential singularities

in the finite part of the plane 214

115. Uniform doubly-periodic functions, without essential singularities in

the finite part of the plane 218

116. Properties of uniform doubly-periodic functions 219

CONTENTS

§§

117. The zeros and the singularities of the derivative of a doubly-periodic

function of the second order . . 231

118, 119. Kelations between homoperiodic functions . . ... . • 233

CHAPTER XL

DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER.

120 121. Formation of an uneven function with two distinct irreducible in
finities; its addition-theorem 243

122, 123. Properties of Weierstrass's o-function . . . . • • 247
124. Introduction of f (2) and of Q(z) 250

125, 126. Periodicity of the function #> (z), with a single irreducible infinity of

degree two; the differential equation satisfied by the function #> (2) 251

127. Pseudo-periodicity of f(«) • 255

128. Construction of a doubly-periodic function in terms of f (z) and its

derivatives • . - . • • 256

129. On the relation qw'- j/eo = ±%iri 25>7

130. Pseudo-periodicity of a (z) • 259

131. Construction of a doubly-periodic function as a product of o-functions ;

with examples 259

132. On derivatives of periodic functions with regard to the invariants

#2 and £3 * ' ' lfK

133 135. Formation of an even function of either class 266

CHAPTER XII.

PSEUDO-PERIODIC FUNCTIONS.

136. Three kinds of pseudo-periodic functions, with the characteristic equa
tions 273

137, 138. Hermite's and Mittag-Leffler's expressions for doubly-periodic functions

of the second kind 275

139. The zeros and the infinities of a secondary function . . - . . 280

140, 141. Solution of Lamp's differential equation 281

142. The zeros and the infinities of a tertiary function .... 286

143. Product-expression for a tertiary function 287

144—146. Two classes of tertiary functions ; Appell's expressions for a function

of each class as a sum of elements 288

147. Expansion in trigonometrical series 293

148. Examples of other classes of pseudo-periodic functions . . . 295

CONTENTS Xvii

CHAPTER XIII.

FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM.
§§ PAGE

149. Definition of an algebraical addition-theorem 297

150. A function defined by an algebraical equation, the coefficients of

which are algebraical functions, or simply-periodic functions, or
doubly-periodic functions, has an algebraical addition-theorem . 297

151 — 154. A function possessing an algebraical addition-theorem is either
algebraical, simply-periodic or doubly-periodic, having in each
instance only a finite number of values for an argument . . 300

155, 156. A function with an algebraical addition-theorem can be defined by a
differential equation of the first order, into which the independent
variable does not explicitly enter 309

CHAPTER XIV.

CONNECTIVITY OF SURFACES.

157—159. Definitions of connection, simple connection, multiple connection, cross
cut, loop-cut . . . .'.•-., 312

160. Relations between cross-cuts and connectivity 315

161. Relations between loop-cuts and connectivity 320

162. Effect of a slit .321

163, 164. Relations between boundaries and connectivity 322

165. Lhuilier's theorem on the division of a connected surface into

curvilinear polygons 325

166. Definitions of circuit, reducible, irreducible, simple, multiple, compound,

reconcileable 327

167, 168. Properties of a complete system of irreducible simple circuits on a

surface, in its relation to the connectivity 328

169. Deformation of surfaces 332

170. Conditions of equivalence for representation of the variable . . 333

CHAPTER XV.

RIEMANN'S SURFACES.

171. Character and general description of a Riemann's surface . - .. •. 336

172. Riemann's surface associated with an algebraical equation . . .338

173. Sheets of the surface are connected along lines, called branch-lines . 338

174. Properties of branch-lines 340

175, 176. Formation of system of branch-lines for a surface ; with examples . 341

177. Spherical form of Riemann's surface . . . 34(5

XV111 CONTENTS

§§ PAGE

178. The connectivity of a Eiemann's surface 347

179. Irreducible circuits : examples of resolution of Riemann's surfaces

into surfaces that are simply connected 350

180, 181. General resolution of a Riemann's surface 353

182. A Riemann's %-sheeted surface when all the branch-points are simple 355

183, 184. On loops, and their deformation 356

185. Simple cycles of Clebsch and Gordan 359

186 — 189. Canonical form of Riemann's surface when all the branch -points are

simple, deduced from theorems of Luroth and Clebsch. . . 361

190. Deformation of the surface 365

191. Remark on uniform algebraical transformations 367

CHAPTER XVI.

ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS.

192. Two subjects of investigation 368

193, 194. Determination of the most general uniform function of position on a

Riemann's surface .......... 369

195. Preliminary lemmas in integration on a Riemann's surface . . . 372

196, 197. Moduli of periodicity for cross-cuts in the resolved surface . . . 373

198. The number of linearly independent moduli of periodicity is equal to

the number of cross-cuts, which are necessary for the resolution

of the surface into one that is simply connected .... 378

199. Periodic functions on a Riemann's surface, with examples . . . 379

200. Integral of the most general uniform function of position on a

Riemann's surface . . . ; 387

201. Integrals, everywhere finite on the surface, connected with the equa

tion w*=S(z) 388

202 — 204. Infinities of integrals on the surface connected with the algebraical
equation f (w, z) = 0, when the equation is geometrically interpret-
able as the equation of a (generalised) curve of the nth order . 388

205, 206. Integrals of the first kind connected with/(w, z) = 0, Demg functions
that are everywhere finite : the number of such integrals, linearly
independent of one another : they are multiform functions . . 394

207, 208. Integrals of the second kind connected with f (w, z) = 0, being func
tions that have only algebraical infinities; elementary integral of
the second kind .......... 396

209. Integrals of the third kind connected with/(w, z) = 0, being functions

that have logarithmic infinities 400

210, 211. An integral of the third kind cannot have less than two logarithmic

infinities ; elementary integral of the third kind .... 401

CONTENTS

CHAPTER XVII.

SCHWARZ'S PROOF OF THE EXISTENCE-THEOREM.

§§ PAGE
212, 213. Existence of functions on a Riemann's surface; initial limitation of

the problem to the real parts u of the functions . ... . . 405

214. Conditions to which u, the potential function, is subject . . . 407

215. Methods of proof : summary of Schwarz's investigation . . . 408
216 — 220. The potential-function u is uniquely determined for a circle by the gene
ral conditions and by the assignment of finite boundary values . 410

221. Also for any plane area, on which the area of a circle can be con-

formally represented 423

222. Also for any plane area which can be obtained by a topological com

bination of areas, having a common part and each conformally
representable on the area of a circle 425

223. Also for any area on a Riemann's surface in which a branch-point

occurs ; and for any simply connected surface .... 428
224 — 227. Real functions exist on a Riemann's surface, everywhere finite, and

having arbitrarily assigned real moduli of periodicity . . . 430

228. And the number of the linearly independent real functions thus ob

tained is 2p ........... 434

229. Real functions exist with assigned infinities on the surface and

assigned real moduli of periodicity. Classes of functions of the
complex variable proved to exist on the Riemann's surface . . 435

CHAPTER XVIII.

APPLICATIONS OF THE EXISTENCE-THEOREM.

230. Three special kinds of functions on a Riemann's surface . . . 437
231 — 233. Relations between moduli of functions of the first kind and those of

functions of the second kind 439

234. The number of linearly independent functions of the first kind on a

Riemann's surface of connectivity 2/; + l is p

235. Normal functions of the first kind ; properties of their moduli .

236. Normal elementary functions of the second kind : their moduli .
237, 238. Normal elementary functions of the third kind : their moduli : inter
change of arguments and parametric points 449

239. The inversion-problem for functions of the first' kind .... 453

240. Algebraical functions on a Riemann's surface without infinities at the

branch-points but only at isolated ordinary points on the surface :
Riemann-Roch's theorem : the smallest number of singularities
that such functions may possess . . . . . . .457

241. A class of algebraic functions infinite only at branch-points . . 460

242. Fundamental equation associated with an assigned Riemann's surface 462

XX CONTENTS

§§ PAGE

243. Appell's factorial functions on a Riemann's surface : their multipliers

at the cross-cuts 464

244, 245. Expression for a factorial function with assigned zeros and assigned
infinities; relations between zeros and infinities of a factorial
function . . ... . . .' . • • • 466

246. Functions defined by differential equations of the form / ( w, -y- ) = 0 470

\ **&)

247 — 249. Conditions that the function should be a uniform function of z. . 471
250, 251. Classes of uniform functions that can be so defined, with criteria of

discrimination . . • , • • • • • • • • 476

(dw\s
~T~ ) =/ (w) .... 482
az j

CHAPTER XIX.

CONFORMAL REPRESENTATION : INTRODUCTORY.

253. A relation between complex variables is the most general relation that

secures conformal similarity between two surfaces .... 491

254. One of the surfaces for conformal representation may, without loss of

generality, be taken to be a plane 495

255, 256. Application to surfaces of revolution ; in particular, to a sphere, so

as to obtain maps .......... 496

257. Some examples of conformal representation of plane areas, in par

ticular, of areas that can be conformally represented on the area

of a circle 501

258. Linear homographic transformations (or substitutions) of the form

w = ,: their fundamental properties 512

cz + d

259. Parabolic, elliptic, hyperbolic and loxodromic substitutions . . . 517

260. An elliptic substitution is either periodic or infinitesimal : substitu

tions of the other classes are neither periodic nor infinitesimal . 521

261. A linear substitution can be regarded geometrically as the result of

an even number of successive inversions of a point with regard

to circles . ......... 523

CHAPTER XX.

CONFORMAL REPRESENTATION : GENERAL THEORY.

262. Riemann's theorem on the conformal representation of a given area

upon the area of a circle with unique correspondence . . . 525

263, 264. Proof of Riemann's theorem : how far the functional equation is

algebraically determinate 526

265, 266. The method of Beltrami and Cayley for the construction of the

functional equation for an analytical curve 530

CONTENTS XXI

§§ PAGE

267, 268. Conformal representation of a convex rectilinear polygon upon the

half-plane of the variable 537

269. The triangle, and the quadrilateral, conformally represented . . 543

270. A convex curve, as a limiting case of a polygon .... 548
271, 272. Conformal representation of a convex figure, bounded by circular arcs :

the functional relation is connected with a linear differential
equation of the second order ........ 549

273. Conformal representation of a crescent ....... 554

274 — 276. Conformal representation of a triangle, bounded by circular arcs . 555
277 — 279. Relation between the triangle, bounded by circles, and the stereographic

projection of regular solids inscribed in a sphere .... 563

280. On families of plane algebraical curves, determined as potential-curves
by a single parameter u + vi : the forms of functional relation

), which give rise to such curves .... 575

CHAPTER XXI.

GROUPS OF LINEAR SUBSTITUTIONS.

281. The algebra of group-symbols 582

282. Groups, which are considered, are discontinuous and have a finite

number of fundamental substitutions 584

283, 284. Anharmonic group : group for the modular-functions, and division of

the plane of the variable to represent the group .... 586
285, 286. Fuchsian groups : division of plane into convex curvilinear polygons :

polygon of reference 591

287. Cycles of angular points in a curvilinear polygon .... 595

288, 289. Character of the division of the plane : example .... 599

290. Fuchsian groups which conserve a fundamental circle . . . 602

291. Essential singularities of a group, and of the automorphic functions

determined by the group ........ 605

292, 293. Families of groups : and their class 606

294. Kleinian groups : the generalised equations connecting two points in

space "... . . .609

295. Division of plane and division of space, in connection with Kleinian

groups 613

296. Example of improperly discontinuous group 615

CHAPTER XXII.

AUTOMORPHIC FUNCTIONS.

297. Definition of automorphic functions . . . . . '- . .619

298. Examples of functions, automorphic for finite discrete groups of sub

stitutions 620

299. Cayley's analytical relation between stereographic projections of posi

tions of a point on a rotated sphere 620

XX11 CONTENTS

§§ PAGE

300. Polyhedral groups ; in particular, the dihedral group, and the tetra-

hedral group 623

301, 302. The tetrahedral functions, and the dihedral functions . . . 628

303. Special illustrations of infinite discrete groups, from the elliptic

modular-functions 633

304. Division of the plane, and properties of the fundamental polygon of

reference, for any infinite discrete group that conserves a funda
mental circle ........... 637

305, 306. Construction of Thetafuchsian functions, pseudo-automorphic for an

infinite group of substitutions 641

307. Relations between the number of irreducible zeros and the number

of irreducible poles of a pseudo-automorphic function, constructed

with a rational algebraical meromorphic function as element . 645

308. Construction of automorphic functions ....... 650

309. The number of irreducible points, for which an automorphic function

acquires an assigned value, is independent of the value . . 651

310. Algebraical relations between functions, automorphic for a group :

application of Riemanu's theory of functions .... 653

311. Connection between automorphic functions and linear differential

equations ; with illustrations from elliptic modular-functions . 654

GLOSSARY OF TECHNICAL TERMS . . . . ; . . ' , . 659
INDEX OF AUTHORS QUOTED . . . . - . . . . . 662
GENERAL INDEX 664

CHAPTER I.

GENERAL INTRODUCTION.

1. ALGEBRAICAL operations are either direct or inverse. Without
entering into a general discussion of the nature of irrational and of imaginary
quantities, it will be sufficient to point out that direct algebraical operations
on numbers that are positive and integral lead to numbers of the same
character; and that inverse algebraical operations on numbers that are
positive and integral lead to numbers, which may be negative or fractional
or irrational, or to numbers which may not even fall within the class of real
quantities. The simplest case of occurrence of a quantity, which is not
real, is that which arises when the square root of a negative quantity is
required.

Combinations of the various kinds of quantities that may occur are of
the form x + iy, where x and y are real and i, the non-real element of the
quantity, denotes the square root of - 1. It is found that, when quantities
of this character are subjected to algebraical operations, they always lead to
quantities of the same formal character; and it is therefore inferred that
the most general form of algebraical quantity is x + iy.

Such a quantity ic + iy, for brevity denoted by z, is usually called a
complex variable*; it therefore appears that the complex variable is the
most general form of algebraical quantity which obeys the fundamental laws
of ordinary algebra.

2. The most general complex variable is that, in which the constituents
x and y are independent of one another and (being real quantities) are
separately capable of assuming all values from - oo to + oo ; thus a doubly-
infinite variation is possible for the variable. In the case of a real variable,
it is convenient to use the customary geometrical representation by measure
ment of distance along a straight line; so also in the case of a complex

* The conjugate complex, viz. x - iy, is frequently denoted by za.
F.

2 GEOMETRICAL REPRESENTATION OF [2.

variable, it is convenient to associate a geometrical representation with
the algebraical expression ; and this is the well-known representation of
the variable ac + iy by means of a point with coordinates x and y referred
to rectangular axes*. The complete variation of the complex variable z
is represented by the aggregate of all possible positions of the associated
point, which is often called the point z ; the special case of real variables
being evidently included in it because, when y = 0, the aggregate of
possible points is the line which is the range of geometrical variation of
the real "variable.

• The variation of z is said to be continuous when the variations of x and y
are contiguous. Continuous variation of z between two given values will
thus be represented by continuous variation in the position of the point z,
that is, by a continuous curve (not necessarily of continuous curvature)
between the points corresponding to the two values. But since an infinite
number of curves can be drawn between two points in a plane, continuity of
line is not sufficient to specify the variation of the complex variable ; and, in
order to indicate any special mode of variation, it is necessary to assign,
either explicitly or implicitly, some determinate law connecting the variations
of x and y or, what is the same thing, some determinate law connecting x
and y. The analytical expression of this law is the equation of the curve
which represents the aggregate of values assumed by the variable between
the two given values.

In such a case the variable is often said to describe the part of the curve
between the two points. In particular, if the variable resume its initial
value, the representative point must return to its initial position ; and then
the variable is said to describe the whole curve -f-.

When a given closed curve is continuously described by the variable,
there are two directions in which the description can take place. From
the analogy of the description of a straight line by a point representing a
real variable, one of these directions is considered as positive and the other

* This method of geometrical representation of imaginary quantities, ordinarily assigned to
Gauss, was originally developed by Argand who, in 1806, published his " Essai sur une maniere
de representer les quantites imaginaires dans les constructions geometriques." This tract was
republished in 1874 as a second edition (Gauthier-Villars) ; an interesting preface is added
to it by Hoiiel, who gives an account of the earlier history of the publications associated with
the theory.

Other references to the historical development are given in Chrystal's Text-book of Algebra,
vol. i, pp. 248, 249; in Holzmiiller's Einfilhrung in die Theorie der isogonalen Venvandschaften
und dcr conformen Abbildungen, verbunden mit Anwendungen auf mathematische Physik, pp. 1 — 10,
21 — 23 ; in Schlomilch's Compendium der hoheren Analysis, vol. ii, p. 38 (note) ; and in Casorati,
Teorica delle funzioni di variabili complesse, only one volume of which was published. In this
connection, an article by Cayley (Quart. Journ. of Math,, vol. xxii, pp. 270 — 308) may be
consulted with advantage.

t In these elementary explanations, it is unnecessary to enter into any discussion of
the effects caused by the occurrence of singularities in the curve.

2-]

THE COMPLEX VARIABLE

Fig. 1.

as negative. The usual convention under which one of the directions is
selected as the positive direction depends upon the conception that the curve
is the boundary, partial or complete, of some area ; under it, that direction is
taken to be positive which is such that the bounded area lies to the left of
the direction of description. It is easy to see that the same direction is taken
to be positive under an equivalent convention
which makes it related to the normal drawn
outwards from the bounded area in the same
way as the positive direction of the axis of y
is to the positive direction of the axis of x
in plane coordinate geometry.

Thus in the figure (fig. 1), the positive
direction of description of the outer curve
for the area included by it is DEF; the
positive direction of description of the inner
curve for the area without it (say, the area
excluded by it) is AGB ; and for the area
between the curves the positive direction of description of the boundary,
which consists of two parts, is DEF, ACB.

3. Since the position of a point in a plane can be determined by means
of polar coordinates, it is convenient in the discussion of complex variables
to introduce two quantities corresponding to polar coordinates.

In the case of the variable z, one of these quantities is (#2 + yn-)l, the
positive sign being always associated with it ; it is called the modulus* of
the variable and it is denoted, sometimes by mod. z, sometimes by \z .

The other is 0, the angular coordinate of the point z ; it is called the
argument (and, less frequently, the amplitude) of the variable. It is
measured in the trigonometrically positive sense, and is determined by
the equations

<K=\Z\ cos 6, y= z\ sin#,

so that z= z\eei. The actual value depends upon the way in which the
variable has acquired its value ; when variation
of the argument is considered, its initial value
is usually taken to lie between 0 and 2?r or, less
frequently, between -TT and +TT.

As z varies in position, the values of \z\
and 6 vary. When z has completed a positive
description of a closed curve, the modulus of z
returns to the initial value whether the origin Fig. 2.

Der absolute Metro,;) is often used by German writers.

1—2

GREAT VALUES OF

[3.

be without, within or on the curve. The argument of z resumes its initial
value, if the origin 0' (fig. 2) be without the curve ; but, if the origin 0 be
within the curve, the value of the argument is increased by 2-rr when z
returns to its initial position.

If the origin be on the curve, the argument of z undergoes an abrupt
change by TT as z passes through the origin ; and the change is an increase
or a decrease according as the variable approaches its limiting position on the
curve from without or from within. No choice need be made between these
alternatives; for care is always exercised to choose curves which do not
introduce this element of doubt.

4. Representation on a plane is obviously more effective for points at a
finite distance from the origin than for points at a very great distance.

One method of meeting the difficulty of representing great values is to
introduce a new variable z1 given by z'z=\\ the part of the new plane for
z which lies quite near the origin corresponds to the part of the old plane
for z which is very distant. The two planes combined give a complete
representation of variation of the complex variable.

Another method, in many ways more advantageous, is as follows. Draw
a sphere of unit diameter, touching the 2-plane at the origin 0 (fig. 3) on
the under side: join a point z in the plane to 0', the other extremity of
the diameter through 0, by a straight line cutting the sphere in Z.
Then Z is a unique representative of z, that is, a single point on the
sphere corresponds to a single point on the plane : and therefore the variable
can be represented on the surface of the sphere. With this mode of

Fig. 3.

representation, 0' evidently corresponds to an infinite value of z : and points
at a very great distance in the 2-plane are represented by points in the
immediate vicinity of 0' on the sphere. The sphei-e thus has the advantage
of putting in evidence a part of the surface jn which the variations of

4.] THE COMPLEX VARIABLE 5

great values of z can be traced*, and of exhibiting the uniqueness of
z — oo as a value of the variable, a fact that is obscured in the represen
tation on a plane.

The former method of representation can be deduced by means of the
sphere. At 0' draw a plane touching the sphere : and let the straight line
OZ cut this plane in z'. Then z is a point uniquely determined by Z
and therefore uniquely determined by z. In this new /-plane take axes
parallel to the axes in the 2-plane.

The points z and / move in the same direction in space round 00'
as an axis. If we make the upper side of the 2-plane correspond to the
lower side of the /-plane, and take the usual positive directions in the
planes, being the positive trigonometrical directions for a spectator looking
at the surface of the plane in which the description takes place, we have
these directions indicated by the arrows at 0 and at 0' respectively, so
that the senses of positive rotations in the two planes are opposite in
space. Now it is evident from the geometry that Oz and O'z' are
parallel ; hence, if 0 be the argument of the point z and & that of the
point z so that 6 is the angle from Ox to Oz and 6' the angle from O'x'
to O'z, we have

6 + ff = ZTT.

Oz 00'
Further, by similar triangles, -^-t = ^-f ,

that is, Oz . O'z' = OO'2 = 1.

Now, if z and z' be the variables, we have

z=0z.eei, z'=0'z'.effi,
so that zz'=0z.0'z' .e^s'^

= 1,
which is the former relation.

The /-plane can therefore be taken as the lower side of a plane touching
the sphere at 0' when the 2-plane is the upper side of a plane touching
it at 0. The part of the 2-plane at a very great distance is represented on
the sphere by the part in the immediate vicinity of 0' : and this part of
the sphere is represented on the /-plane by its portion in the immediate
vicinity of 0', which therefore is a space wherein the variations of infinitely
great values of z can be traced.

But it need hardly be pointed out that any special method of represent
ation of the variable is not essential to the development of the theory of
functions ; and, in particular, the foregoing representation of the variable,
when it has very great values, merely provides a convenient method of
dealing with quantities that tend to become infinite in magnitude.

* This sphere is sometimes called Neumann's sphere; it is used by him for the representation
of the complex variable throughout his treatise Vorlesungen uber Riemann'a Theorie der AlcVschen
Integrate (Leipzig, Teubner, '2nd edition, 1884).

6 CONDITIONS OF [5.

5. The simplest propositions relating to complex variables will be
assumed known. Among these are, the geometrical interpretation of opera
tions such as addition, multiplication, root-extraction ; some of the relations
of complex variables occurring as roots of algebraical equations with real
coefficients; the elementary properties of functions of complex variables
which are algebraical and integral, or exponential, or circular functions;
and simple tests of convergence of infinite series and of infinite products*.

6. All ordinary operations effected on a complex variable lead, as already
remarked, to other complex variables; and any definite quantity, thus
obtained by operations on z, is necessarily a function of z.

But if a complex variable w be given as a complex function of x
and y without any indication of its source, the question as to whether
w is or is not a function of z requires a consideration of the general idea
of functionality.

It is convenient to postulate u + iv as a form of the complex variable w,
where u and v are real. Since w is initially unrestricted in variation, we
may so far regard the quantities u and v as independent and therefore as
any functions of x and y, the elements involved in z. But more explicit
expressions for these functions are neither assigned nor supposed.

The earliest occurrence of the idea of functionality is in connection with
functions of real variables ; and then it is coextensive with the idea of
dependence. Thus, if the value of X depends on that of x and on no other
variable magnitude, it is customary to regard X as a function of x\ and
there is usually an implication that X is derived from x by some series of
operations^.

A detailed knowledge of z determines x and y uniquely ; hence the values
of u and v may be considered as known and therefore also w. Thus the
value of w is dependent on that of z, and is independent of the values
of variables unconnected with z; therefore, with the foregoing view of
functionality, w is a function of z.

It is, however, equally consistent with that view to regard w as a complex
function of the two independent elements from which z is constituted ; and
we are then led merely to the consideration of functions of two real
independent variables with (possibly) imaginary coefficients.

* These and other introductory parts of the subject are discussed in Chrystal's Text-book of
Algebra and in Hobson's Treatise on Plane Trigonometry.

They are also discussed at some length in the recently published translation, by G. L.
Cathcart, of Harnack's Elements of the differential and integral calculus (Williams and Norgate,
1891), the second and the fourth books of which contain developments that should be consulted
in special relation with the first few chapters of the present treatise.

These books, together with Neumann's treatise.cited in the note on p. 5, will hereafter be cited
by the names of their respective authors.

t It is not important for the present purpose to keep in view such mathematical expressions
as have intelligible meanings only when the independent variable is confined within limits.

6.] FUNCTIONAL DEPENDENCE 7

Both of these aspects of the dependence of w on z require that z be
regarded as a composite quantity involving two independent elements which
can be considered separately. Our purpose, however, is to regard z as the
most general form of algebraical variable and therefore as an irresoluble
entity ; so that, as this preliminary requirement in regard to z is unsatisfied,
neither of the aspects can be adopted.

7. Suppose that w is regarded as a function of z in the sense that it can
be constructed by definite operations on z regarded as an irresoluble
magnitude, the quantities u and v arising subsequently to these operations
by the separation of the real and the imaginary parts when z is replaced by
x + iy. It is thereby assumed that one series of operations is sufficient for
the simultaneous construction of u and v, instead of one series for u and
another series for v as in the general case of a complex function in § 6.
If this assumption be justified by the same forms resulting from the two
different methods of construction, it follows that the two series of opera
tions, which lead in the general case to u and to v, must be equivalent to
the single series and must therefore be connected by conditions ; that is, u
and v as functions of a; and y must have their functional forms related.

We thus take

u + iv — w = f(z) = f(x + iy)

without any specification of the form of f. When this postulated equation

is valid, we have

dw dw dz ,. , . dw

_ — . _ _ - I { 2/9 TTT _

dx dz dx J ^ ' dz'
dw _ dw "dz _ .,,. . . dw
frj = ~fad~y~V (Z) lfa'

• dw 1 dw dw

and therefore — = -— = —- ........................... (1)

dx i dy dz

equations from which the functional form has disappeared. Inserting the
value of w, we have

whence, after equating real and imaginary parts,

dv _du du _ dv
dx dy' dx dy"
These are necessary relations between the functional forms of u and v.

These relations are easily seen to be sufficient to ensure the required
functionality. For, on taking w = ii + iv, the equations (2) at once lead to

dw _ 1 dw
dx i dy '

,, , . dw .dw

that is, to -- — \- 1 — - = 0,

ox dy

8 RIEMANN'S [7.

a linear partial differential equation of the first order. To obtain the most
general solution, we form a subsidiary system

dx _ dy _ dw
T==T ==~0~*

It possesses the integrals w, x + iy; and then from the known theory of
such equations we infer that every quantity w satisfying the equation can be
expressed as a function of x + iy, i.e., of z. The conditions (2) are thus
proved to be sufficient, as well as necessary.

8. The preceding determination of the necessary and sufficient conditions
of functional dependence is based upon the existence of a functional form ;
and yet that form is not essential, for, as already remarked, it disappears from
the equations of condition. Now the postulation of such a form is equivalent
to an assumption that the function can be numerically calculated for each
particular value of the independent variable, though the immediate expres
sion of the assumption has disappeared in the present case. Experience of
functions of real variables shews that it is often more convenient to use
their properties than to possess their numerical values. This experience is
confirmed by what has preceded. The essential conditions of functional
dependence are the equations (1), and they express a property of the function

w, viz., that the value of the ratio -r is the same as that of ~- , or, in other

words, it is independent of the manner in which dz ultimately vanishes by
the approach of the point z + dz to coincidence with the point z. We are
thus led to an entirely different definition of functionality, viz. :

A complex quantity w is a function of another complex quantity z, when
they change together in such a manner that the value of -, is independent of
the value of the differential element dz.

This is Riemann's definition* ; we proceed to consider its significance.

We have

dw du + idv
dz dx + idy

/du .dv\ dx /du .dv\ du

__ I __ I n _ I _ _____ I I __ L ^ __ I _ Y. _

~~ \dx dxj dx + idy \dy dy/ dx + idy '
Let </> be the argument of dz ; then

_

cos <£ + 1 sin </>

* Ges. Werke, p. 5; a modified definition is adopted by him, ib., p. 81.

8.] DEFINITION OF A FUNCTION

and therefore

dw . (du .dv .du dv) „,. {du .dv .du dv

I I I i n I I I 1 a—^4> I J I t In

7 — i i«i T" ^ • 5 • 57 f • a** i<5 Tfc ^~ ~r * « «

«£ • [da; dx dy dy} (dx dx dy dy

Since -j— is to be independent of the value of the differential element dz,
dz

it must be independent of <f> the argument of dz ; hence the coefficient
of e-2*« in the preceding expression must vanish, which can happen only if

du _dv dv _ du
dx dy' dx dy "

These are necessary conditions; they are evidently also sufficient to make
^— independent of the value of dz and therefore, by the definition, to secure
that w is a function of z.

By means of the conditions (2), we have

dw _ du .dv _dw
dz dx dx dx '

dw .du dv 1 dw
and also — - = — i - — [_=_.

dz dy dy i dy

agreeing with the former equations (1) and immediately derivable from the
present definition by noticing that dx and idy are possible forms of dz.

It should be remarked that equations (2) are the conditions necessary
and sufficient to ensure that each of the expressions

udx — vdy and vdx + udy

is a perfect differential — a result of great importance in many investigations
in the region of mathematical physics.

When the conditions (2) are expressed, as is sometimes convenient, in
terms of derivatives with regard to the modulus of z, say r, and the
argument of z, say 0, they take the new forms

du_ldv dv _ Idu,

^ — ~ 57j > ^~ — ^TT. (^)-

or r dv or r da

We have so far assumed that the function has a differential coefficient —
an assumption justified in the case of functions which ordinarily occur. But
functions do occur which have different values in different regions of the

.z-plane, and there is then a difficulty in regard to the quantity ,W at the

boundaries of such regions ; and functions do occur which, though themselves
definite in value in a given region, do not possess a differential coefficient at
all points in that region. The consideration of such functions is not of
substantial importance at present : it belongs to another part of our subject.

10 CONFORMAL [8.

It must not be inferred that, because -j- is independent of the direction
in which dz vanishes when w is a function of z, therefore -=- has only one

value. The number of its values is dependent on the number of values of w :
no one of its values is dependent on dz.

A quantity, defined as a function by Riemann on the basis of this
property, is sometimes* called an analytical function; but it seems pre
ferable to reserve the term analytical in order that it may be associated
hereafter (§ 34) with an additional quality of the functions.

9. The geometrical interpretation of complex variability leads to impor
tant results when applied to two variables w and z which are functionally
related.

Let P and p be two points in different planes, or in different parts of
the same plane, representing w and z respectively; and suppose that P and
p are at a finite distance from the points (if any) which cause discontinuity
in the relationship. Let q and r be any two other points, z + dz and z + 8z,
in the immediate vicinity of p ; and let Q and E be the corresponding
points, w + dw and w + &w, in the immediate vicinity of P. Then

dw j ^ dw ?
dw = ^r- dz. bw = -r— of,
dz dz

the value of ~ being the same for both equations, because, as w is a function
dz

of z, that quantity is independent of the differential element of z. Hence

8w _ Bz
dw dz'

on the ground that , is neither zero nor infinite at z, which is assumed not

CL2

to be a point of discontinuity in the relationship. Expressing all the differ
ential elements in terms of their moduli and arguments, let

dz = a-eei, dw — rje^1,

Sz = oV'*, 8w = i)<$\
and let these values be substituted in the foregoing relation ; then

77' tr

tj a

$-$ = &-&.

Hence the triangles QPR and qpr are similar to one another, though
not necessarily similarly situated. Moreover the directions originally chosen
for pq and pr are quite arbitrary. Thus it appears that a functional relation

* Harnack, § 84.

1 11 V - (<M\* a. i^v

• I <a I — I "5 / ' \ ">

$a?/ \oyj \dy

9.] REPRESENTATION OF PLANES 11

between two complex variables establishes the similarity of the corresponding
infinitesimal elements of those parts of two planes which are in the immediate
vicinity of the points representing the two variables.

The magnification of the w-plane relative to the ^-plane at the corre
sponding points P and p is the ratio of two corresponding infinitesimal

lengths, say of QP and qp. This is the modulus of -^— ; if it be denoted by

m, we have

2 _ dw 2
dz

_ du dv du dv
dx dy dy dx '

Evidently the quantity m, in general, depends on the variables and
therefore it changes from one point to another ; hence the functional relation
between w and z does not, in general, establish similarity of finite parts of
the two planes corresponding to one another through the relation.

It is easy to prove that w = az + b, where a and b are constants, is the
only relation which establishes similarity of finite parts ; and that, with this
relation, a must be a real constant in order that the similar parts may be
similarly situated.

If u + iv = w = <}> (z), the curves u = constant and v = constant cut at
right angles; a special case of the proposition that, if <£ (x + iy) = u + v^,
where A, is a real constant and u, v are real, then u= constant and v= constant
cut at an angle X.

The process, which establishes the infinitesimal similarity of two planes
by means of a functional relation between the variables of the planes, may be
called the conformal representation of one plane on another*.

The discussion of detailed questions connected with the conformal representation is
deferred until the later part of the treatise, principally in order to group all such
investigations together ; but the first of the two chapters, devoted to it, need not be
deferred so late and an immediate reading of some portion of it will tend to simplify
many of the explanations relative to functional relations as they occur in the early
chapters of this treatise.

10. The analytical conditions of functionality, under either of the
adopted definitions, are the equations (2). From them it at once follows that

8^ + ty* = '

* By Gauss (Ges. Werke, t. iv, p. 262) it was styled conforme Abbildung, the name
universally adopted by German mathematicians. The French title is representation conforme ;
and, in England, Cayley has used orthomorphosis or ortliomorphic transformation.

12 CONDITIONS OF FUNCTIONAL DEPENDENCE [10.

so that neither the real nor the imaginary part of a complex function can be
arbitrarily assumed.

If either part be given, the other can be deduced ; for example, let u be
given ; then we have

7 j j

dv = ^-dx + — dy

dx dy

du , du j
= -=-dx+~-dy,
dy ox '

and therefore, except as to an additive constant, the value of v is

[i 9w 7 du -, \

- — dx + 5- dy I .
A dy ax ° I

In particular, when u is an integral function, it can be resolved into the
sum of homogeneous parts

MI + w2 + w3 + . . . ;

and then, again except as to an additive constant, v can similarly be
expressed in the form

Vl + V2 + V3 + ----

It is easy to prove that

dum dum

™>» = y-te-*-ty>

by means of which the value of v can be obtained.

The case, when u is homogeneous of zero dimensions, presents no
difficulty ; for we then have

v = c-a\ogr, =c-/f£
where a, 6, c are constants.

Similarly for other special cases; and, in the most general case, only
a quadrature is necessary.

The tests of functional dependence of one complex on another are of
effective importance in the case when the supposed dependent complex
arises in the form u + iv, where u and v are real; the tests are, of course,
superfluous when w is explicitly given as a function of z. When w does
arise in the form u + iv and satisfies the conditions of functionality, perhaps
the simplest method (other than by inspection) of obtaining the explicit
expression in terms of z is to substitute z — iy for x in u + iv ; the simplified
result must be a function of z alone.

11. Conversely, when w is explicitly given as a function of z and it is
divided into its real and its imaginary parts, these parts individually
satisfy the foregoing conditions attaching to u and v. Thus logr, where r
is the distance of a point z from a point a, is the real part of log (z — a)
and therefore satisfies the equation

11.]

EXAMPLE OF RIEMANN S DEFINITION

13

Again, <f>, the angular coordinate of z relative to the same point a, is
the real part of — i log (z — a) and satisfies the same equation : the more
usual form of <£ being tan"1 {(y — y0)/(® — %o)}> where a = x0 + iy0. Again, if
a point z be distant r from a and r' from b, then log (r/r'\ being the real
part of log {(z — a)l(z — b)\, is a solution of the same equation.

The following example, the result of which will be useful subsequently*, uses the
property that the value of the derivative is independent of the differential element.

z-c

Consider a function

u + iv = w = log

where c' is the inverse of c with regard to a circle centre the origin 0 and radius R.
Then

z-c

* V

:— r>

z-c

and the curves u = constant are circles. Let

W- •
(fig. 4) Oc = r, xOc = a so that c = reat, c'= — eal;

then if

Fig. 4.

the values of X for points in the interior of the circle of radius R vary from zero, when
circle u = constant is the point c, to unity, when the circle u = constant is the circle of
radius R. Let the point K ( = 6eal) be the centre of the circle determined by a value of
X, and let its radius be p ( = %MN}. Then since

cM r ,. cN

we have

whence

r+p-d r d + P~r

— Vp-B Q-p

r r

P =

Now if dn be an element of the normal drawn inwards at z to the circle NzM, we have
dz = dx+idy= — dn . cos ^ - idn . sin ^

--«*<*»,

where ^ ( = zKx'} is the argument of z relative to the centre of the circle. Hence, since

dw 1 1

we have
But

so that

and

, ., , du .dv

and therefore -=- + i -j- =
dn dn

__ _

dz z — c z-c'1
du .dv dw
dn

.dv dw /I 1 \ ty

dn dn \z — c' z — c)

e^ - Reai) •

J> _ 1 _ 1 !_

I /i! ~\ ^^ 7? *1^ X ff */^
\ A7*6 — J\G ./t6 i

* In § 217, in connection with the investigations of Schwarz, by whom the result is stated,
Ges. Werke, t. ii, p. 183.

14 DEFINITIONS [11.

Hence, equating the real parts, it follows that

du (_R2-r2A2)2

dn ~ \R(R*- r2) {E2 - 2Rr\ cos (^ - Q) + XV2} '
the differential element dn being drawn inwards from the circumference of the circle.

The application of this method is evidently effective when the curves u = constant,
arising from a functional expression of w in terms of z, are a family of non-intersecting
algebraical curves.

12. As the tests which are sufficient and necessary to ensure that a
complex quantity is a function of z have been given, we shall assume that
all complex quantities dealt with are functions of the complex variable
(§§ 6, 7). Their characteristic properties, their classification, and some of
the simpler applications will be considered in the succeeding chapters.

Some initial definitions and explanations will now be given.

(i). It has been assumed that the function considered has a differential
coefficient, that is, that the rate of variation of the function in any direction
is independent of that direction by being independent of the mode of change
of the variable. We have already decided (§ 8) not to use the term analytical
for such a function. It is often called monogenic, when it is necessary to
assign a specific name ; but for the most part we shall omit the name, the
property being tacitly assumed*.

We can at once prove from the definition that, when the derivative

/ dw\ •.-.'•, if- c <-• v dw Idw

w, = -p- exists, it is itselt a Junction, .bor w-, =-=— = - =— are equations
\ dz ) dx i dy

which, when satisfied, ensure the existence of w^ ; hence

1 dw-! _ 1 3 (dw\
i dy i dy \d% )
_ d_ (I dw\
dx \i dyj
_dw1
= l)x '

shewing, as in § 8, that the derivative ~ is independent of the direction in

CL2

which dz vanishes. Hence wl is a function of z.

Similarly for all the derivatives in succession.

(ii). Since the functional dependence of a complex is ensured only if the
value of the derivative of that complex be independent of the manner in
which the point z + dz approaches to coincidence with z, a question naturally

* This is in fact done by Biemann, who calls such a dependent complex simply a function.
Weierstrass, however, has proved (§ 85) that the idea of a monogenic function of a complex
variable and the idea of dependence expressible by arithmetical operations are not coextensive.
The definition is thus necessary; but the practice indicated in the text will be adopted, as non-
monogenic functions will be of relatively rare occurrence.

12.] DEFINITIONS 15

suggests itself as to the effect on the character of the function that may be
caused by the manner in which the variable itself has come to the value of z.
If a function have only one value for each given value of the variable,
whatever be the manner in which the variable has come to that value, the
function is called uniform*. Hence two different paths from a point a to a
point z give at z the same value for any uniform function ; and a closed
curve, beginning at any point and completely described by the ^-variable,
will lead to the initial value of w, the corresponding w-curve being closed, if z
have passed through no point which makes w infinite.

The simplest class of uniform functions is constituted by algebraical
rational functions.

(iii). If a function have more than one value for any given value of the
variable, or if its value can be changed by modifying the path in which
the variable reaches that given value, the function is called multiform-]'.
Characteristics of curves, which are graphs of multiform functions corre
sponding to a 2-curve, will hereafter be discussed.

One of the simplest classes of multiform functions is constituted by
algebraical irrational functions.

(iv). A multiform function has a number of different values for the same
value of z, and these values vary with z : the aggregate of the variations of
any one of the values is called a branch of the function. Although the
function is multiform for unrestricted variation of the variable, it often
happens that a branch is uniform when the variable is restricted to
particular regions in the plane.

(v). A point in the plane, at which two or more branches of a multiform
function assume the same value, is called a branch-point^ of the function;
the relations of the branches in the immediate vicinity of a branch-point will
hereafter be discussed.

(vi). A function which is monogenic, uniform and continuous over any
part of the ^-plane is called holomorphic § over that part of the plane. When
•a function is called holomorphic without any limitation, the usual implication
is that the character is preserved over the whole of the plane which is not at
infinity.

The simplest example of a holomorphic function is a rational integral
algebraical polynomial.

* Also monodromic, or monotropic; with German writers the title is eindeutig, occasionally,
einandrig.

t Also polytropic ; with German writers the title is mchrdeittig.

J Also critical point, which, however, is sometimes used to include all special points of a
function ; with German writers the title is Verziveigungspunkt, and sometimes Windungspunkt.
French writers use point de ramification, and Italians punto di giramento and punto di
diramazione.

§ Also synectic.

16 EXAMPLES ILLUSTRATING [12.

(vii). A root (or a zero) of a function is a value of the variable for which
the function vanishes.

The simplest case of occurrence of roots is in a rational integral alge
braical function, various theorems relating to which (e.g., the number of
roots included within a given contour) will be found in treatises on the
theory of equations.

(viii). The infinities of a function are the points at which the value of
the function is infinite. Among them, the simplest are the poles* of the
function, a pole being an infinity such that in its immediate vicinity the
reciprocal of the function is holomorphic.

Infinities other than poles (and also the poles) are called the singular
points of the function : their classification must be deferred until after the
discussion of properties of functions.

(ix). A function which is monogenic, uniform and, except at poles,
continuous, is called a meromorphic function f. The simplest example is a
rational algebraical fraction.

13. The following functions give illustrations of some of the preceding
definitions.

(a) In the case of a meromorphic function

F(z)
111 — — * — -

/<*)'

where F and / are rational algebraical functions without a common factor,
the roots are the roots of F (z) and the poles are the roots of f (z). Moreover,
according as the degree of F is greater or is less than that of f,z = vo is a
pole or a zero of w.

(b) If w be a polynomial of order n, then each simple root of w is a

branch-point and a zero of wm, where m is a positive integer ; z = oo is
a pole of w; and z= oo is a pole but not a branch-point or is an infinity
(though not a pole) and a branch-point of w$ according as n is even or odd.

(c) In the case of the function

1

w-

sn-

z

(the notation being that of Jacobian elliptic functions), the zeros are given by

z
for all positive and negative integral values of m and of m'. If we take

- = iK' + 2mK + Zm'iK' -f £

z

* Also polar discontinuities ; also (§ 32) accidental singularities.

t Sometimes rey-nlar, but this term will be reserved for the description of another property of
functions.

13.] THE DEFINITIONS 17

where £ may be restricted to values that are not large, then

w = (- l)m &sn£

so that, in the neighbourhood of a zero, w behaves like a holomorphic
function. There is evidently a doubly-infinite system of zeros: they are
distinct from one another except at the origin, where an infinite number
practically coincide.

The infinities of w are given by

for all positive and negative integral values of n and of n'. If we take

- = 2nK + Zn'iK' + £

2!

then - = (-l)"sn£

w

so that, in the immediate vicinity of f=0, - is a holomorphic function.

Hence f = 0 is a pole of w. There is thus evidently a doubly-infinite system
of poles ; they are distinct from one another except at the origin, where an
infinite number practically coincide. But the origin is not a pole; the
function, in fact, is there not determinate, for it has an infinite number of
zeros and an infinite number of infinities, and the variations of value are not
necessarily exhausted.

For the function — j , the origin is a point which will hereafter be called

sn-

z

an essential singularity.

F.

CHAPTER II.

INTEGRATION OF UNIFORM FUNCTIONS.

14. THE definition of an integral, that is adopted when the variables
are complex, is the natural generalisation of that definition for real variables
in which it is regarded as the limit of the sum of an infinite number of
infinitesimally small terms. It is as follows : —

Let a and z be any two points in the plane ; and let them be connected
by a curve of specified form, which is to be the path of variation of the
independent variable. Let f(z) denote any function of 0; if any infinity
of f(z) lie in the vicinity of the curve, the line of the curve will be chosen
so as not to pass through that infinity. On the curve, let any number of
points z^ z2,..., zn in succession be taken between a and z ; then, if the sum

(z, - a)f (a) + (z, - z,} f (z,) + ... + (z- zn)f(zn}

have a limit, when n is indefinitely increased so that the infinitely numerous
points are in indefinitely close succession along the whole of the curve from a
to z, that limit is called the integral of / (z) between a and z. It is denoted,
as in the case of real variables, by

f(z)dz.

The limit, as the value of the integral, is associated with a particular
curve : in order that the integral may have a definite value, the curve (called
the path of integration) must, in the first instance, be specified*. The
integral of any function whatever may not be assumed to depend in general
only upon the limits.

15. Some inferences can be made from the definition.
(I.) The integral along any path from a to z passing through a point £ is
the sum of the integrals from a to £ and from \ to z along the same path.

* This specification is tacitly supplied when the variables are real : the variable point moves
along the axis of x.

15.] INTEGRATION 19

Analytically, this is expressed by the equation

P / (*) dz = I V (*) dz + I V (*) <fc,

^ a J a J f

the paths on the right-hand side combining to form the path on the left.

(II.) When the path is described in the reverse direction, the sign of the
integral is changed : that is,

the curve of variation between a and z being the same.

(III.) The integral of the sum of a finite number of terms is equal to
the sum of the integrals of the separate terms, the path of integration being
the same for all.

(IV.) If a function f (z) be finite and continuous along any finite line
between two points a and z, the integral \ f(z)dz is finite.

J a

Let 7 denote the integral, so that we have I as the limit of

r=0

hence |/| = limit of

Because f(z} is finite and continuous, its modulus is finite and therefore
must have a superior limit, say M, for points on the line. Thus

80 that I/I < limit of r+1

<MS,

where 8 is the finite length of the path of integration. Hence the modulus
of the integral is finite ; the integral itself is therefore finite.

No limitation has been assigned to the path, except finiteness in length ;
the proposition is still true when the curve is a closed curve of finite length.

Hermite and Darboux have given an expression for the integral which
leads to the same result. We have as above

f(z)\ dz\,
where 6 is a real positive quantity less than unity. The last integral involves

2—2

20 THEOREMS [15.

only real variables; hence* for some point £ lying between a and z, we have

f

J a

so that l/| = fl9f|/(!)|.

It therefore follows that there is some argument a such that, if X = Be10-,

This form proves the finiteness of the integral ; and the result is the
generalisation f to complex variables of the theorem just quoted for real
variables.

(V.) When a, function is expressed in the form of a series, which converges
uniformly and unconditionally, the integral of the function along any path of
finite length is the sum of the integrals of the terms of the series along the
same path, provided that path lies within the circle of convergence of the series :
— a result, which is an extension of (III.) above.

Let M0 + MI + u.2 + . . . be the converging series ; take

/ (z) = U0 + M! + . . . + Un + R,

where \R\ can be made infinitesimally small with indefinite increase of n,
because the series converges uniformly and unconditionally. Then by (III.),
or immediately from the definition of the integral, we have

rz rs rz rz re

f(z)dz= I u0dz + ^dz + . .. + I undz + 1 Rdz,

J a J a J a J a J a

the path of integration being the same for all the integrals. Hence, if

re n re

(S) = I f (z) dz — 2 I umdz,

J a m=oJ a

ft

we have © = I Rdz.

ft

= I

J a

Let R be the greatest value of \R\ for points in the path of integration
from a to z, and let 8 be the length of this path, so that 8 is finite ;

then, by (IV.),

\®\<SR.

Now 8 is finite ; and, as n is increased indefinitely, the quantity R tends
towards zero as a limit for all points within the circle of convergence and
therefore for all points on the path of integration provided that the path lie
within the circle of convergence. When this proviso is satisfied, |@| becomes
infinitesimally small and therefore also ® becomes infinitesimally small with

* Todhunter's Integral Calculus (4th ed.), § 40; Williamson's Integral Calculus, (Gth ed.), § 96.
t Hermite, Cours d la faculte dcs sciences de Paris (46mc ed., 1891), p. 59, where the reference
to Darboux is given.

15.]

ON INTEGRATION

21

indefinite increase of n. Hence, under the conditions stated in the enuncia
tion, we have

rs oo r%

f(z)dz- 2 I umdz = 0,

J a m^QJ a

which proves the proposition.

16. The following lemma* is of fundamental importance.

Let any region of the plane, on which the ^-variable is represented, be
bounded by one or more simple^ curves which do not meet one another:
each curve that lies entirely in the finite part of the plane will be considered
to be a closed curve.

If ' p and q be any two functions of cc and y, which, for all points within the
region or along its boundary, are uniform, finite and continuous, then the
integral

fffdq dp\j ,
1 1 a - a dxdy,

JJ \dx dyj

extended over the whole area of the region, is equal to the integral

f(pdx + qdy),
taken in a positive direction round the whole boundary of the region.

(As the proof of the proposition does not depend on any special form of
region, we shall take the area to be (fig. 5) that which is included by the
curve QiPiQs'Pa' and excluded by P^Qz'PsQs and excluded by P/P2. The
positive directions of description of the curves are indicated by the arrows ;
and for integration in the area the positive directions are those of increas
ing a; and increasing y.)

AB

Fig. 5.

* It is proved by Eiemann, Ges. Werke, p. 12, and is made by him (as also by Cauchy) the
basis of certain theorems relating to functions of complex variables.

t A curve is called simple, if it have no multiple points. The aim, in constituting the boundary
from such curves is to prevent the superfluous complexity that arises from duplication of area on
the plane. If, in any particular case, multiple points existed, the method of meeting the difficulty
would be to take each simple loop as a boundary.

22 FUNDAMENTAL THEOREM [16.

First, suppose that both p and q are real. Then, integrating with regard
to x, we have *

where the brackets imply that the limits are to be introduced. When the
limits are introduced along a parallel GQ^... to the axis of x, then, since
CQiQi'. • • gives the direction of integration, we have

[qdy] = - qjdyj. + qi'dt/i - q.2dy2 + q-2'dy2' - q3dy3 + q»dy9',

where the various differential elements are the projections on the axis of y
of the various elements of the boundary at points along GQiQJ....

Now when integration is taken in the positive direction round the whole
boundary, the part of / qdy arising from the elements of the boundary at the
points on CQjQ/... is the foregoing sum. For at Q3' it is qa'dy3 because the
positive element dy9, which is equal to CD, is in the positive direction of
boundary integration; at Q3 it is —q3dys because the positive element dy3,
also equal to CD. is in the negative direction of boundary integration ;
at Qz it is q2'dy2', for similar reasons ; at Q.2 it is — q2dya, for similar reasons ;
and so on. Hence

corresponding to parallels through C and D to the axis of x, is equal to
the part of fqdy taken along the boundary in the positive direction for all
the elements of the boundary that lie between those parallels. Then when
we integrate for all the elements CD by forming f[qdy], an equivalent is
given by the aggregate of all the parts of fqdy taken in the positive direction
round the whole boundary ; and therefore

on the suppositions stated in the enunciation.
Again, integrating with regard to y, we have

when the limits are introduced along a parallel RP^P^. . . to the axis of y :
the various differential elements are the projections on the axis of x of the
various elements of the boundary at points along SPjP/....

It is proved, in the same way as before, that the part of - jpdx arising
from the positively-described elements of the boundary at the points on
BP^'... is the foregoing sum. At P3 the part of fpdac is - p3'dx3, because
the positive element dx3, which is equal to AB, is in the negative direction

* It is in this integration, and in the corresponding integration for p, that the properties of
the function q are assumed : any deviation from uniformity, finiteness or continuity within the
region of integration would render necessary some equation different from the one given in
the text.

16.] IN INTEGRATION 23

of boundary integration ; at P3 it is p3dx3, because the positive element
dx3, also equal to AB, is in the positive direction of boundary integration;
and so on for the other terms. Hence

- [pdas],

corresponding to parallels through A and B to the axis of y, is equal to
the part of fpdx taken along the boundary in the positive direction for all
the elements of the boundary that lie between those parallels. Hence
integrating for all the elements AB, we have as before

[[dp j j , j

~ dxdy = — I pax,

JJdy

and therefore II U ?r ) dxdy=f(pdx + qdy).

Secondly, suppose that p and q are complex. When they are resolved
into real and imaginary parts, in the forms p' + ip" and q' + iq" respectively,
then the conditions as to uniformity, finiteness and continuity, which apply to
p and q, apply also to p', q', p", q". Hence

and ~ - - dxdy = j(p"dx + q"dy),

and therefore 1 1 [ 2* _ J9 j dxdy = J(pdx + qdy}

JJ \ox oy/

which proves the proposition.

No restriction on the properties of the functions p and q at points
that lie without the region is imposed by the proposition. They may have
infinities outside, they may cease to be continuous at outside points or they
may have branch-points outside ; but so long as they are finite and continuous
everywhere inside, and in passing from one point to another always acquire
at that other the same value whatever be the path of passage in the region,
that is, so long as they are uniform in the region, the lemma is valid.

17. The following theorem due to Cauchy* can now be proved : _
If a function f(z) be holomorphic throughout any region of the z-plane,
then the integral ff(z) dz, taken round the whole boundary of that region, is zero.
We apply the preceding result by assuming

p=f(z\ q = ip = if(z);

owing to the character of f(z), these suppositions are consistent with the

* For an account of the gradual development of the theory and, in particular, for a
statement of Cauchy's contributions to the theory (with references), see Casorati, Teorica
delle funzioni di variabili complcsse, pp. 64-90, 102-106. The general theory of functions,
as developed by Briot and Bouquet in their treatise Theoric des fonctiom ellipUques, is based
upon Cauchy's method.

24 INTEGRATION OF [17.

conditions under which the lemma is valid. Since p is a function of z, we
have, at every point of the region,

dp _ I dp

das i dy '
and therefore, in the present case,

dq _ . dp _ dp

das doc dy '

There is no discontinuity or infinity of p or q within the region ; hence

the integral being extended over the region. Hence also

!(pdx + qdy) = 0, A^ ^/
when the integral is taken round the whole boundary of the region. But

pdx + qdy = pdx + ipdy
— pdz
=f(z)dz,

and therefore //(X) dz = 0,

the integral being taken round the whole boundary of the region within
which f(z) is holomorphic.

It should be noted that the theorem requires no limitation on the cha
racter of/(^) for points z that are not included in the region.

Some important propositions can be derived by means of the theorem, as
follows.

18. When a function f (z) is holomorphic over any continuous region

rz
of the plane, the integral I f(z)dz is a holomorphic function of 2 provided the

J a

points z and a as well as the whole path of integration lie within that region.

The general definition (§ 14) of an integral is associated with a specified
path of integration. In order to prove that the integral is a holomorphic
function of z, it will be necessary to prove (i) that the integral acquires the
same value in whatever way the point z is attained, that is, that the value is
independent of the path of integration, (ii) that it is finite, (iii) that it
is continuous, and (iv) that it is monogenic.

Let two paths ayz and afiz between a and z be drawn (fig. 6) in the
continuous region of the plane within which f(z) is
holomorphic. The line ayzfia is a contour over the area
of which / (z} is holomorphic ; and therefore ff(z) dz
vanishes when the integral is taken along ayzfta.
Dividing the integral into two parts and implying by
Zy, Zp that the point z has been reached by the paths a"
a<yz, a{3z respectively, we have Fig. 6.

18.] HOLOMOEPHIC FUNCTIONS 25

and therefore */ (z) dz = - f (z) dz

J a J Zg

-?/*/(*)*

J a
Thus the value of the integral is independent of the way in which z has

FZ

acquired its value ; and therefore I f(z) dz is uniform in the region. Denote
it by F(z).

Secondly, f(z) is finite for all points in the region and, after the result
of § 17, we naturally consider only such paths between a and z as are finite in
length, the distance between a and z being finite; hence (§ 15, IV.) the
integral F (z} is finite for all points z in the region.

Thirdly, let z' (= z 4- 82) be a point infinitesimally near to z ; and consider
I f(z) dz. By what has just been proved, the path from a to z' can be taken

J d

aftzz' ; therefore

(*/(*) dz = [/(z) dz + lZf(z) dz

J * J a J z

fz+8z rz rz+Sz

or f(z}dz- \ f(z)dz=\ f(z)dz,

J a J a J z

fz+Sz

80 that F(z + Sz) - F(z) = f(z} dz.

J 2

Now at points in the infinitesimal line from z to z' , the value of the
continuous function f(z) differs only by an infinitesimal quantity from its
value at z ; hence the right-hand side is

where e| is an infinitesimal quantity vanishing with ck It therefore follows
that

is an infinitesimal quantity with a modulus of the same order of small
quantities as \Sz\. Hence F (z) is continuous for points z in the region.
Lastly, we have

and therefore F(z + Sz)-F(z)

82

has a limit when Sz vanishes; and this limit, f(z), is independent of the
way in which 8z vanishes. Hence F (z) has a differential coefficient ; the
integral is monogenic for points z in the region.

26 INTEGRATION OF [18.

Hence F (z), which is equal to

* f(z)d*t

is uniform, finite, continuous and monogenic; it is therefore a holomorphic
function of z.

As in § 16 for the functions p and q, so here for f(z), no restriction is
placed on properties of / (z) at points that do not lie within the region; so
that elsewhere it may have infinities, or discontinuities or branch points.
The properties, essential to secure the validity of the proposition, are
(i) that no infinities or discontinuities lie within the region, and (ii) that the
same value of f(z) is acquired by whatever path in the continuous region
the variable reaches its position z.

COROLLARY. No change is caused in the value of the integral of a
holomorphic function between two points when the path of integration between
the points is deformed in any manner, provided only that, during the defor
mation, no part of the path passes outside the boundary of the region within
which the function is holomorphic.

This result is of importance, because it permits special forms of the path
of integration without affecting the value of the integral.

19. When a function f(z) is holomorphic over a part of the plane
bounded by two simple curves (one lying within the other), equal values of
ff(z) dz are obtained by integrating round each of the curves in a direction,
which — relative to the area enclosed by each — is positive.

The ring-formed portion of the plane (fig. 1, p. 3) which lies between
the two curves being a region over which f(z) is holomorphic, the integral
ff(z) dz taken in the positive sense round the whole of the boundary of
the included portion is zero. The integral consists of two parts : first, that
round the outer boundary the positive sense of which is DEF', and second,
that round the inner boundary the positive sense of which for the portion of
area between ABC and DEF is ACE. Denoting the value of ff(z)dz round
DEF by (DEF), and similarly for the other, we have

(ACB) + (DEF) = 0.

The direction of an integral can be reversed if its sign be changed, so that
(ACB) = - (ABC) ; and therefore

(ABC) = (DEF).

But (ABC) is the integral ff(z)dz taken round ABC, that is, round the
curve in a direction which, relative to the area enclosed by it, is positive.

The proposition is therefore proved.

The remarks made in the preceding case as to the freedom from limitations
on the character of the function outside the portion are valid also in this case.

19.] HOLOMORPHIC FUNCTIONS 27

COROLLARY I. When the integral of a function is taken round the whole
of any simple curve in the plane, no change is caused in its value by continuously
deforming the curve into any other simple curve provided that the function
is holomorphic over the part of the plane in which the deformation is effected.

COROLLARY II. When a function f (z) is holomorphic over a continuous
portion of a plane bounded by any number of simple non-intersecting curves,
all but one of which are external to one another and the remaining one of
which encloses them all, the value of the integral jf(z) dz taken positively round
the single external curve is equal to the sum of the values taken round each of
the other curves in a direction which is positive relative to the area enclosed
by it.

These corollaries are of importance in finding the value of the integral
of a meromorphic function round a curve which encloses one or more of the
poles. The fundamental theorem for such integrals, also due to Cauchy, is
the following.

20. Let f(z) denote a function which is holomorphic over any region in
the z-plane and let a denote any point within that region, which is not a zero

°ff(2); then

., , 1 f/0) ,

f(a) = ^— • *-*-* az>

2vnJ z-a

the integral being taken positively round the whole boundary of the region.

With a as centre and a very small radius p, describe a circle G, which will
be assumed to lie wholly within the region; this assumption is justifiable
because the point a lies within the region. Because f (z) is holomorphic over
the assigned region, the f unction f(z)l(z — a) is holomorphic over the whole of
the region excluded by the small circle C. Hence, by Corollary II. of § 19, we
have

z-a

the notation implying that the integrations are taken round the whole
boundary B and round the circumference of G respectively.

For points on the circle C, let z — a = peei, so that 9 is the variable for
the circumference and its range is from 0 to 2?r ; then we have

dz

z — a

= id6.

Along the circle f(z)=f (a + peei) ; the quantity p is very small and / is
finite and continuous over the whole of the region so that f(a + peei) differs
from /(«) only by a quantity which vanishes with p. Let this difference
be e, which is a continuous small quantity; then |ej is a small quantity
which, for every point on the circumference of C, vanishes with p. Then

28 INTEGRATION OF [20.

" edO.
o

If E denote the value of the integral on the right-hand side, and 77 the
greatest value of the modulus of e along the circle, then, as in § 15,

/•2ir

i E < I e d6

f

Now let the radius of the circle diminish to zero: then 77 also diminishes
to zero and therefore E , necessarily positive, becomes less than any finite
quantity however small, that is, E is itself zero; and thus we have

z — a
which proves the theorem.

This result is the simplest case of the integral of a meromorphic

f(z}
function. The subject of integration is — — , a function which is monogenic

and uniform throughout the region and which, everywhere except at z = a, is
finite and continuous ; moreover, z = a is a pole, because in the immediate

Z ~~~ CL

vicinity of a the reciprocal of the subject of integration, viz. ^-rr > i-B h°l°-
morphic.

The theorem may therefore be expressed as follows :

If g (z) be a meromorphic function, which in the vicinity of a can be

f(z}
expressed in the form J where f(a) is not zero and which at all other

Z — CL

points in a region enclosing a is holomorphic, then

- — . fg (z) dz = limit of (z — a)g (z) when z — a,

the integral being taken round a curve in the region enclosing the point a.

The pole a of the function g (z) is said to be simple, or of the first order,
or of multiplicity unity.

Corollary. The more general case of a meromorphic function with a
finite number of poles can easily be deduced. Let these be a1}..., an each
assumed to be simple ; and let

G (z) = (z- a,) (z - aa). ..(z - an).

20.] MEROMORPHIC FUNCTIONS 29

Let f(z) be a holomorphic function within a region of the 2-plane bounded
by a simple contour enclosing the n points a1} a»,...an, no one of which is a
zero off(z). Then since

f(z) » 1 f(z)

we have j^~( = S „,, . -^-- .

6r (#) r=i Or (ar) z — ar

w ^ f u f/(*),j 3 ! f/(*) ,7

We therefore have "L , ' dz = 2< >.. , . I dz,

J &(*) r=iCr (ar)J 2-ar

each integral being taken round the boundary. But the preceding proposition
gives

because f(z) is holomorphic over the whole region included in the contour ;
and therefore

the integral on the left-hand side being taken in the positive direction*.

The result just obtained expresses the integral of the meromorphic
function round a contour which includes a finite number of its simple poles.
It can be otherwise obtained by means of Corollary II. of § 19, by adopting
a process similar to that adopted above, viz., by making each of the curves in
the Corollary quoted small circles round the points Oj,..., an with ultimately
vanishing radii.

21. The preceding theorems have sufficed to evaluate the integral of
a function with a number of simple poles : we now proceed to obtain
further theorems, which can be used among other purposes to evaluate
the integral of a function with poles of order higher than the first.

We still consider a function f(z) which is holomorphic within a given
region. Then, if a be a point within the region which is not a zero of f(z),
we have

z - a

the point a being neither on the boundary nor within an infinitesimal
distance of it. Let a + Sa be any other point within the region ; then

dz,

z — a — 8a

* We shall for the future assume that, if no direction for a complete integral be specified, the
positive direction is taken.

30

and therefore

PROPERTIES OF

[21.

iff,

8a

f(z)dz,

t J ((* - a)2 (z - of (z-a -Sa)j
the integral being in every case taken round the boundary.

Since f(z) is monogenic, the definition of /'(a), the first derivative of
/(a), gives /'(a) as the limit of

f(a + Ba)-f(a)
Ba

when Ba ultimately vanishes ; hence we may take

where a is a quantity which vanishes with Ba and is therefore such that \ a \
also vanishes with Ba. Hence

dividing out by Sa and transposing, we have

As yet, there is no limitation on the value of Sa ; we now proceed to a limit
by making a + Ba approach to coincidence with a, viz., by making Ba
ultimately vanish. Taking moduli of each of the members of the last
equation, we have

(a) _ i f J(*

2in j (z - o

_„ + ««.

(z — a)2 (z — a — Ba)

27T

dz

Let the greatest modulus of -. ~ =r—. for points z along the

(j — a)2 (z — a — Ba)

boundary be M, which is a finite quantity on account of the conditions
applying to f(z) and the fact that the points a and a + Ba are not
infinitesimally near the boundary. Then, by § 15,

t

dz

'0-a)2 (z-a-Ba)

<MS,

where 8 is the whole length of the boundary, a finite quantity. Hence

1 f f(z} , , |8a|

dz

c

ITT

21.] HOLOMORPHIC FUNCTIONS 31

When we proceed to the limit in which Sa vanishes, we have Ba = 0
and |o-| = 0, ultimately; hence the modulus on the left-hand side ultimately
vanishes and therefore the quantity to which that modulus belongs is itself
zero, that is,

,

(z — of

so that / (a) = —-. !/-^~n dz.

ZTTI )(z- of

This theorem evidently corresponds in complex variables to the well-known
theorem of differentiation with respect to a constant under the integral
sign when all the quantities concerned are real.

Proceeding in the same way, we can prove that

/ (a + &*)-/ (a) _ 2!_ f /(*)
Ba ~2Trij(z-af

where 6 is a small quantity which vanishes with Ba. Moreover the integral
on the right-hand side is finite, for the subject of integration is everywhere
finite along the path of integration which itself is of finite length. Hence,
first, a small change in the independent variable leads to a change of the
same order of small quantities in the value of the function f (a), which
shews that f (a) is a continuous function. Secondly, denoting

&*) -/(a)

by &/'(«), we have the limiting value of -— *— - equal to the integral on

the right-hand side when Sa vanishes, that is, the derivative of f (a) has
a value independent of the form of 8a and therefore /' (a) is monogenic.
Denoting this derivative by /"(a), we have

J (z — a)3

Thirdly, the function f (a) is uniform ; for it is the limit of the value
of — - -- x-- — J-\J and both /(a) and /(a + Sa) are uniform. Lastly, it

is finite; for (S 15) it is the value of the integral - — . l.^—^dz, in which

2?n J (z — af

the length of the path is finite and the subject of integration is finite at
every point of the path.

Hence f (a) is continuous, monogenic, uniform, and finite throughout
the whole of the region in which f (z) has these properties: it is a
holomorphic function. Hence : —

When a function is holomorphic in any region of the plane bounded

32 PROPERTIES OF [21.

by a simple curve, its derivative is also holomorphic within that region. And,
by repeated application of this theorem : —

When a function is holomorphic in any region of the plane bounded
by a simple curve, it has an unlimited number of successive derivatives each
of which is- holomorphic within the region.

All these properties have been shewn to depend simply upon the holo
morphic character of the fundamental function ; but the inferences relating
to the derivatives have been proved only for points within the region and
not for points on the boundary. If the foregoing methods be used to prove
them for points on the boundary, they require that a consecutive point shall
be taken in any direction ; in the absence of knowledge about the fundamental
function for points outside (even though just outside) no inferences can be
justifiably drawn.

An illustration of this statement is furnished by the hypergeometric series
which, together with all its derivatives, is holomorphic within a circle of
radius unity and centre the origin ; and the series converges unconditionally
everywhere on the circumference, provided 7 > a. + /3. But the corresponding
condition for convergence on the circumference ceases to be satisfied for some
one of the derivatives and for all which succeed it : as such functions do not
then converge unconditionally, the circumference of the circle must be
excluded from the region within which the derivatives are holomorphic.

22. Expressions for the first and the second derivatives have been
obtained.

By a process similar to that which gives the value of f (a), the derivative
of order n is obtainable in the form

n ' f f (z\

/<») (a) = — . I, ' dz,
J w 2wt J (z - a)n+l

the integral being taken round the whole boundary of the region or round any
curves which arise from deformation of the boundary, provided that no point
of the curves in the final or any intermediate form is indefinitely near to a.

In the case when the curve of integration is a circle, no point of which
circle may lie outside the boundary of the region, we have a modified form
fcr /*'(•>

For points along the circumference of the circle with centre a and radius

r, let

z — a = reei,

dz

so that as before — = idO :

z — a

then 0 and 2?r being taken as the limits of 0, we have

22.] HOLOMORPHIC FUNCTIONS 33

Let M be the greatest value of the modulus of f (z) for points on the
circumference (or, as it may be convenient to consider, of points on or within
the circumference) : then

\f(n)(a)\<~ e-nei\\f(a

i / \ / 1 ^ 27ryw * -

nl

<

M

Now, let there be a function <£ (s) defined by the equation

M

— a

which can evidently be expanded in a series of ascending powers of z — a
that converges within the circle. The series is

-

[dnd> (z)~\ , M

Hence —!L±J =n\ —

[ d*» ]z=a *>•

so that, if the value of the nth derivative of $(z), when z = a, be denoted
by <£<n> (a), we have

|/»(a)| «p>(a).

These results can be extended to functions of more than one variable :
the proof is similar to the foregoing proof. When the variables are two,
say z and z', the results may be stated as follows : —

^ For all points z within a given simple curve 0 in the ^-plane and all
points / within a given simple curve G' in the /-plane, let / (z, z) be a
holomorphic function; then, if a be any point within C and a' any point
within G',

^n+nJ (a, a')

J (z - a)n+1 (z' — aTf

where n and ri are any integers and the integral is taken positively round the
two curves G and G'.

If M be the greatest value of \f (z, z'} for points z and z within their
respective regions when the curves G and G' are circles of radii r, r' and
centres a, a', then

dn+n'f(a, a') M

~3aW»' <w!/i!rv^5

F.

34 HOLOMORPHIC FUNCTIONS [22-

M
and if $(?>*)

dn+n'f(a,a')

dn+n'(j> (z, z')

then da»da'«

when z = a and z = a' in the derivative of <£ (z, z).

23. All the integrals of meromorphic functions that have been considered
have been taken along complete curves : it is necessary to refer to integrals
along curves which are lines only from one point to another. A single
illustration will suffice at present.

Consider the integral f -t-^-dz; the function /» is

J H0 z — a

supposed holomorphic in the given region, and z and z0 are
any two points in that region. Let some curves joining z
to z0 be drawn as in the figure (fig. 7).

~ , •* 2o

is holomorphic over the whole area en- Fig> 7

z— a

closed by z^zSz0: and therefore we have ^ = 0 when taken round the

boundary of that area. Hence as in the earlier case we have

z — a Jz0 z — a
The point a lies within the area enclosed by z0yz^z0, and the function

is holomorphic, except in the immediate vicinity of z = a ; hence

r f ( v\

I - dz = 2Trif(a),

J z — a

the integral on the left-hand side being taken round Z0yzj3z0. Hence

z — a

Denoting ^-by g(z), the function g (z) has one pole a in the region

£ "~ CL

considered.

The preceding results are connected only with the simplest form of
meromorphic functions; other simple results can be derived by means of the
other theorems proved in §§ 17—21. Those which have been obtained are
sufficient however to shew that : The integral of a meromorphic function
fg(z)dz from one point to another of the region of the function is not in
general a uniform function. The value of the integral is not altered by
any deformation of the path which does not meet or cross a pole of the
function; but the value is altered when the path of integration is so

23.] GENERAL PROPOSITIONS IN INTEGRATION 35

deformed as to pass over one or more poles. Therefore it is necessary to
specify the path of integration when the subject of integration is a mero-
morphic function ; only partial deformations of the path of integration are
possible without modifying the value of the integral.

24. The following additional propositions* are deduced from limiting
cases of integration round complete curves. In the first, the curve becomes
indefinitely small ; in the second, it becomes infinitely large. And in neither,
are the properties of the functions to be integrated limited as in the pre
ceding propositions, so that the results are of wider application.

I. If f(z) be a function which, whatever be its character at a, has no
infinities and no branch-points in the immediate vicinity of a, the value of
ff(z)dz taken round a small circle with its centre at a tends towards zero
when the circle diminishes in magnitude so as ultimately to be merely the
point a, provided that, as z — a diminishes indefinitely, the limit of (z — a)f(z)
tend uniformly to zero.

Along the small circle, initially taken to be of radius r, let

*-a-fl*i *

dz

so that = idO,

z— a

and therefore Sf(z) dz = i\ (z — a)f(z) d6.

Jo

Hence \ff(z)dz\ = I *" (z - a)f(z) d0

Jo

<r\(z-a)f(z)\de
Jo

rzn

< Md0
Jo

where M' is the greatest value of M, the modulus of (z - a)f(z), for points
on the circumference. Since (z - a)f(z) tends uniformly to the limit zero as
| z -a diminishes indefinitely, \jf(z) dz\ is ultimately zero. Hence the integral
itself jf(z)dz is zero, under the assigned conditions.

Note. If the integral be extended over only part of the circumference of
the circle, it is easy to see that, under the conditions of the proposition,
the value offf(z)de still tends towards zero.

COROLLARY. If (z-a)f(z) tend uniformly to a limit k as \z-a\
diminishes indefinitely, the value of ff(z)dz taken round a small circle centre
a tends towards 27rik in the limit.

* The form of the first two propositions, which is adopted here, is due to Jordan, Cours
d' Analyse, t. ii, §§ 285, 286.

3—2

36 GENERAL PROPOSITIONS [24.

Thus the value of [- dz j, taken round a very small circle centre «, where a is

~ d * /2V

not the origin, is zero : the value of f - - - -, round the same circle is -. ( - \ .

J (a — z) (a-M)

Neither the theorem nor the corollary will apply to a function, such as sn —-^

which has the point a for an essential singularity: the value of (z-a)sn^— ^, as
\z-a\ diminishes indefinitely, does not tend (§ 13) to a uniform limit. As a matter of

fact the function sn — has an infinite number of poles in the immediate vicinity of a
z- a

as the limit z—a, is being reached.

II. Whatever be the character of a function f (z} for infinitely large values
ofz, the value ofjf(z) dz, taken round a circle with the origin for centre, tends
towards zero as the circle becomes infinitely large, provided that, as \z\
increases indefinitely, the limit of zf(z) tend uniformly to zero.

Along a circle, centre the origin and radius R, we have z =Eeei, so that

dz .ja
- = idd,

z

r-2ir

and therefore // 0) dz = i zf(z) d6.

Jo

Hence I //(*)<&! = £*

<T zf(z)\dS
Jo

rzn

< Mde

Jo

<

where M' is the greatest value of M, the modulus of zf(z)t for points on
the circumference. When R increases indefinitely, the value of M' is zero
on the hypothesis in the proposition; hence \$f(*)d*\ is ultimately zero.
Therefore the value of ff(z) dz tends towards zero, under the assigned con
ditions.

Note. If the integral be extended along only a portion of the circumfer
ence, the value of jf(z}dz still tends towards zero.

COROLLARY. // zf(z) tend uniformly to a limit k as \z . increases
indefinitely, the value of jf(z) dz, taken round a very large circle, centre the
origin, tends towards %7rik.

Thus the value of J(l -zn}~^dz round an infinitely large circle, centre the origin, is zero
if n > 2, and is 2ir if » = 2.

III. If all the infinities and the branch-points of a function lie in a finite
region of the z-plane, then the value of jf(z) dz round any simple curve, which

24.] IN INTEGRATION 37

includes all those points, is zero, provided the value of zf(z\ as \z\ increases
indefinitely, tends uniformly to zero.

The simple curve can be deformed continuously into the infinite circle
of the preceding proposition, without passing over any infinity or any
branch- point ; hence, if we assume that the function exists all over the plane,
the value of jf(z) dz is, by Cor. I. of § 19, equal to the value of the integral
round the infinite circle, that is, by the preceding proposition, to zero.

Another method of stating the proof of the theorem is to consider
the corresponding simple curve on Neumann's sphere (§ 4). The surface
of the sphere is divided into two portions by the curve*: in one portion lie
all the singularities and the branch-points, and in the other portion there is
no critical point whatever. Hence in this second portion the function is holo-
morphic ; since the area is bounded by the curve we see that, on passing back
to the plane, the excluded area is one over which the function is holomorphic.
Hence, by § 19, the integral round the curve is equal to the integral round
an infinite circle having its centre at the origin and is therefore zero, as
before.

COROLLARY. If, under the same circumstances, the value of zf(z}, as
\z increases indefinitely, tend uniformly to k, then the value of $f(z)dz round
the simple curve is

Thus the value of I — — r along any simple curve which encloses the two points
J (a2 - z2)*

a and - a is 2ir ; the value of

dz

{(!-«") (!-*%•)}*

round any simple curve enclosing the four points 1, -1, T, -7, is zero, k being a non-

1C K

vanishing constant ; and the value of J(l — z2n)~*dz, taken round a circle, centre the origin
and radius greater than unity, is zero when n is an integer greater than 1.

/dz
~ ~ — 771

K*-«i) (*-««)(*-«•)}*

round any circle, which has the origin for centre and includes the three distinct points
€lt e2, e3, is not zero. The subject of integration has 2 = 00 for a branch-point, so that the
condition in the proposition is not satisfied ; and the reason that the result is no longer
valid is that the deformation into an infinite circle, as described in Cor. I. of § 19,
is not possible because the infinite circle would meet the branch-point at infinity.

25. The further consideration of integrals of functions, that do not possess
the character of uniformity over the whole area included by the curve of in
tegration, will be deferred until Chap. ix. Some examples of the theorems
proved in the present chapter will now be given.

* The fact that a single path of integration is the boundary of two portions of the surface
of the sphere, within which the function may have different characteristic properties, will be
used hereafter (§ 104) to obtain a relation between the two integrals that arise according as the
path is deformed within one portion or within the other.

38 EXAMPLES IN [25.

Ex. 1. It is sufficient merely to mention the indefinite integrals (that is, integrals from
an arbitrary point to a point z} of rational, integral, algebraical functions. After the
preceding explanations it is evident that they follow the same laws as integrals of similar
functions of real variables.

/dz
,— ^ , taken round a simple curve.

When n is 0, the value of the integral is zero if the curve do not include the point a,
and it is Ziri if the curve include the point a.

When n is a positive integer, the value of the integral is zero if the curve do not
include the point a (by § 17), and the value of the integral is still zero if the curve do
include the point a (by § 22, for the function f(z) of the text is 1 and all its derivatives
are zero). Hence the value of the integral round any curve, which does not pass through
a, is zero.

We can now at once deduce, by § 20, the result that, if a holomorphic function be
constant along any simple closed curve within its region, it is constant over the whole
area within the curve. For let t be any point within the curve, z any point on it, and C
the constant value of the function for all the points z ; then

B'

mn

2 — t

the integral being taken round the curve, so that

<&M-— t dz

= C
by the above result, since the point t lies within the curve.

Ex. 3. Consider the integral \e~^dz.

In any finite part of the plane, the function e~02 is holomorphic; therefore (§ 17) the
integral round the boundary of a rectangle
(fig. 8), bounded by the lines x= ±a, y = 0,
y=b, is zero : and this boundary can be
extended, provided the deformation remain
in the region where the function is holo
morphic. Now as a tends towards infinity,
the modulus of e~z\ being e~x2 + y2, tends
towards zero when y remains finite ; and
therefore the preceding rectangle can be Fig. 8.

extended towards infinity in the direction of the axis of x, the side b of the rectangle
remaining unaltered.

Along A' A, we have z=x : so that the value of the integral along the part A' A of the

fa

boundary is I e~x dx.

J -a

Along AB, we have z = a + iy, so that the value of the integral along the part AB

f*

is i I e~(a + iyrdy.

Jo
Along BB', we have z = x + ib, so that the value of the integral along the part BB'

f'a
is I e-(x + lVdx.

J a

Along B'A', we have z=-a + iy, so that the value of the integral along the part

B'A1 is i (V(-«H
J t,

25.] INTEGRATION 39

/•ft
The second of these portions of the integral is e~a<i . » . I tP~***tefy, which is easily seen

J o
to be zero when the (real) quantity a is infinite.

Similarly the fourth of these portions is zero.

Hence as the complete integral is zero, we have, on passing to the limit,

I e~^dx+\ e-^2ibx + b'2da;=0,

J -<*> J oo

whence e62 I e~ *-***&?=* I

J -oo J -<*

/oo
e'3^ (cos 2bx—i sin

and therefore, on equating real parts, we obtain the well-known result

/

J -Q

This is only one of numerous examples* in which the theorems in the text can be
applied to obtain the values of definite integrals with real limits and real variables.

rzn-i
Ex. 4. Consider the integral I - --- dz. where n is a real positive quantity less than

J 1+z
unity.

The only infinities of the subject of integration are the origin and the point - 1 ;
the branch-points are the origin and 2=00. Everywhere else in the plane the function
behaves like a holomorphic function ; and, therefore, when we take any simple closed
curve enclosing neither the origin nor the point — 1, the integral of the function round
that curve is zero.

We shall assume that the curve lies on the positive side of the axis of x and that it
is made up of : —

(i) a semicircle (73 (fig. 9), centre the origin and radius R which is made to increase
indefinitely :

Fig. 9.

(ii) two semicircles, ct and c2, with their centres at 0 and — 1 respectively, and with

radii r and /, which ultimately are made infinitesimally small :
(iii) the diameter of (73 along the axis of x excepting those ultimately infinitesimal

portions which are the diameters of cx and of c2.

The subject of integration is uniform within the area thus enclosed although it
is not uniform over the whole plane. We shall take that value of zn~l which has its
argument equal to (n— 1) 6, where 6 is the argument of z.

* See Briot and Bouquet, Theorie des fonctions elliptiques, (2nd ed.), pp. 141 et sqq., from
which examples 3 and 4 are taken.

40 EXAMPLES IN [25.

The integral round the boundary is made up of four parts.

0H — 1

(a) The integral round (73. The value of z . , as z \ increases indefinitely, tends

uniformly to the limit zero ; hence, as the radius of the semicircle is increased indefinitely,
the integral round (73 vanishes (§ 24, n., Note).

^n— 1

(b) The integral round cv The value of z . , as | z \ diminishes indefinitely,

1 -\-z

tends uniformly to the limit zero ; hence as the radius of the semicircle is diminished
indefinitely, the integral round cv vanishes (§ 24, I., Note}.

zn-l

(c) The integral round c2. The value of (1 + 2) , as |1+2| diminishes indefinitely

A ~r z

for points in the area, tends uniformly to the limit (— I)""1, i.e., to the limit g(M~1)'™.
Hence this part of the integral is

being taken in the direction indicated by the arrow round c2) the infinitesimal semicircle.
Evidently -- =id6 and the limits are TT to 0, so that this part of the whole integral is

idd

(d) The integral along the axis of x. The parts at — 1 and at 0 which form the
diameters of the small semicircles are to be omitted ; so that the value is

-l+r' J r
This is what Cauchy calls the principal value* of the integral

/"°° /yH ~ 1

/ •* 7

I dx.

Since the whole integral is zero, we have

ineniri+ I Y — dx = 0.

Let P = I ^ — dx, P' = I dx,

and 0— I dx,

J o 1-a?

principal values being taken in each case. Then, taking account of the arguments, we have

Since iwenvi + P + 1* = 0,

we have P - eH7riQ = - inenni,

* Williamson's Integral Calculus, % 104.

25.] INTEGRATION 41

so that P— Q cos nn — ir sin nn, Q sin ntr = TT cos nir.

Hence I ; dx—P = ir cosec TOTT,

jo 1+a?

dx—Q = ir cot %TT.

. 5. In the same way it may be proved that

.

where n is an integer, a is positive and o> is e*2" .

Jik 6. By considering the integral Je-2^™-1^ round the contour of the sector of a
circle of radius r, bounded by the radii 0=0, 6=a, where a is less than |TT and n is positive,
it may be proved that

„»— 1 „-;

{r';
on proceeding to the limit when r is made infinite. (Briot and Bouquet.)

Ex. 7. Consider the integral I ~~^, where n is an integer. The subject of integration

is meromorphic ; it has for its poles (each of which is simple) the n points o>r for r=0,
1, ..., n-l, where a is a primitive nth root of unity ; and it has no other infinities and no

branch -points. Moreover the value of — — -, as \z\ increases indefinitely, tends uniformly

to the limit zero ; hence (§ 24, in.) the value of the integral, taken round a circle centre
the origin and radius > 1, is zero.

This result can be derived by means of Corollary II. in § 19. Surround each of the
poles with an infinitesimal circle having the pole for centre ; then the integral round
the circle of radius > 1 is equal to the sum of the values of the integral round the
infinitesimal circles. The value round the circle having «r for its centre is, by § 20,

2rri( limit of " , when z = u>r}
\ z - L J

Hence the integral round the large circle

2

n r=n

= 0.

Ex. 8. Hitherto, in all the examples considered, the poles that have occurred have
been simple : but the results proved in § 21 enable us to obtain the integrals of
functions which have multiple poles within an area. As an example, consider the

integral / (1+g2)n + i round any curve which includes the point i but not the point - i, these
points being the two poles of the subject of integration, each of multiplicity n + l.

42 EXAMPLES IN INTEGRATION [25.

We have seen that /" (a) = j^ J ^_a^n + i *»

where /(«) is holomorphic throughout the region bounded by the curve round which the
integral is taken.

In the present case a is i, and f(z) = . «.n ^\ ', s° that

2» ! (-l)n

u. <• /*n - 2-

and therefore /" (*J = ^j (2i)»*-n "~ ~ wT

Hence we have "***™'

In the case of the integral of a function round a simple curve which contains several
of its poles we first (§ 20) resolve the integral into the sum of the integrals round simple
curves each containing only one of the points, and then determine each of the latter
integrals as above.

Another method that is sometimes possible makes use of the expression of the uniform
function in partial fractions. After Ex. 2, we need retain only those fractions which are of

the form — : the integral of such a fraction is ZniA, and the value of the whole integral

z-a

is therefore tor&A. It is thus sufficient to obtain the coefficients of the inverse first powers
which arise when the function is expressed in partial fractions corresponding to each pole.
Such a coefficient A, the coefficient of -j in the expansion of the function, is called by

Z (Jj

Cauchy the residue of the function relative to the point.
For example,

so that the residues relative to the points -1, -o>, -to2 are f, £«, |«2 respectively.
Hence if we take a semicircle, of radius > 1 and centre the origin with its diameter
along the axis of y, so as to lie on the positive side of the axis of y, the area between the
semi-circumference and the diameter includes the two points -« and -«2 ; and therefore

the value of

dz

taken along the semi-circumference and the diameter, is

&*&»+!•?);

i.e., the value is - *ni.

CHAPTER III.

EXPANSION OF FUNCTIONS IN SERIES OF POWERS.

26. WE are now in a position to obtain the two fundamental theorems
relating to the expansion of functions in series of powers of the variable :
they are due to Cauchy and Laurent respectively.

Cauchy 's theorem is as follows*: —

When a function is holomorphic over the area of a circle of centre a, it can
be expanded as a series of positive integral powers of z-a converging for all
points within the circle.

Let z be any point within the circle; describe a concentric circle of
radius r such that

\z-a\ = p <r<R, ^ ^i.

where R is the radius of the given circle. If t
denote a current point on the circumference of the
new circle, we have

dt

t — a z — a
t — a

Fif?. 10.

the integral extending along the whole circumference of radius r. Now

z-a

t-a

z -an+l

z — a

— a

t—a

so that, by § 14 (III.), we have

J_ f
27ri]

f(t)

t-z\t-a

dt.

* Exercices d' 'Analyse et de Physique Mathe'matiqne, t. ii, pp. 50 et seq. ; the memoir was first
made public at Turin in 1832.

44

CAUCHY'S THEOREM ON THE [26.

Now /(«) is holomorphic over the whole area of the circle ; hence, if t be
not actually on the boundary of the region (§§ 21, 22), a condition secured by
the hypothesis r < R, we have

and therefore

(z-a)n (z-a)n+l

"

Let the last term be denoted by L. Since z — a =p and \t-a\ = r,
it is at once evident that \t-z\^r-p. Let M be the greatest value of
|/(0| for points along the circle of radius r ; then M must be finite, owing to
the initial hypothesis relating tof(z). Taking

f — n — TP6i

v W/ ~~ I C*

so that dt = i(t- a) d6,

P«+> t*m de

we have \L\ = -f

i

^P

Jlf

rn (i — p)

Jffl-:C

\r)
Now r was chosen to be greater than p ; hence as n becomes infinitely

large, we have W infinitesimally small. Also If (1 — ?l is finite.
\r/ V f/

Hence as ?i increases indefinitely, the limit of |i|, necessarily not negative,
is infinitesimally small and therefore, in the same case, L tends towards
zero.

It thus appears, exactly as in § 15 (V.), that, when n is made to increase
without limit, the difference between the quantity f(z) and the first n + 1
terms of the series is ultimately zero ; hence the series is a converging series
having f(z) as the limit of the sum, so that

which proves the proposition under the assigned conditions. It is the form
of Taylor's expansion for complex variables.

Note. The series on the right-hand side is frequently denoted by
P(z — a), where P is a general symbol for a converging series of positive
integral powers of z — a: it is also sometimes* denoted by P(z\a). Con-

* Weierstrass, Abh. am der Functionenlehre, p. 1.

26.] EXPANSION OF A FUNCTION 45

formably with this notation, a series of negative integral powers of z — a

would be denoted by P I - — ) ; a series of negative integral powers of z

\z — a/

either by P (-) or by P(^|oo), the latter implying a series proceeding in
\zj

positive integral powers of a quantity which vanishes when z is infinite,
i.e., in positive integral powers of — .

Z

If, however, the circle can be made of infinitely great radius so that the
function f(z) is holomorphic over the finite part of the plane, the equivalent
series is denoted by G(z — a) and it converges over the whole plane.
Conformably with this notation, a series of negative integral powers of z - a

which converges over the whole plane is denoted by G I - - j .

27. The following remarks on the proof and on inferences from it should
be noticed.

(i) In order that — - - may be expanded in the required form, the
t — z

point z must be taken actually within the area of the circle of radius R ;
and therefore the convergence of the series P (z — a) is not established for
points on the circumference.

(ii) The coefficients of the powers of z — a in the series are the
values of the function and its derivatives at the centre of the circle ; and the
character of the derivatives is sufficiently ensured (§ 21) by the holomorphic
character of the function for all points within the region. It therefore
follows that, if a function be holomorphic within a region bounded by a
circle of centre a, its expansion in a series of ascending powers of z — a
converging for all points within the circle depends only upon the values of
the function and its derivatives at the centre.

But instead of having the values of the function and of all its derivatives
at the centre of the circle, it will suffice to have the values of the holomorphic
function itself over any small region at a or along any small line through
a, the region or the line not being infinitesimal. The values of the
derivatives at a can be found in either case ; for /' (b) is the limit of
{f(b + 86) —f(b)}/8b, so that the value of the first derivative can be found
for any point in the region or on the line, as the case may be ; and so for all
the derivatives in succession.

(iii) The form of Maclaurin's series for complex variables is at once
derivable by supposing the centre of the circle at the origin. We then
infer that, if a function be kolomorphic over a circle, centre tJie origin, it can be

46 DARBOUX'S EXPRESSION [27.

represented in the form of a series of ascending, positive, integral powers of the
variable given by

where the coefficients of the various powers of z are the values of the derivatives
of f(z) at the origin, and the series converges for all points within the circle.

Thus, the function ez is holomorphic over the finite part of the plane ;
therefore its expansion is of the form G (z). The function log (1 4- z) has a
singularity at — 1 ; hence within a circle, centre the origin and radius unity,
it can be expanded in the form of an ascending series of positive integral
powers of z, it being convenient to choose that one of the values of the
function which is zero at the origin. Again, tan"1.?2 has singularities at the
four points z4 = — I, which all lie on the circumference; choosing the value at
the origin which is zero there, we have a similar expansion in a series, con
verging for points within the circle.

Similarly for the function (1 +z)n, which has — 1 for a singularity.

(iv) Darboux's method* of derivation of the expansion of f (z) in
positive powers of z — a depends upon the expression, obtained in § 15 (IV.),
for the value of an integral. When applied to the general term

1 Uz-a\n+i s,.. ,,
f(t)dt,

= L say, it gives L = \r fe^J /(f),

where £ is some point on the circumference of the circle of radius r, and X is

2 ~ fl

a complex quantity of modulus not greater than unity. The modulus of ^

b ~~ a

is less than a quantity which is less than unity ; the terms of the series of
moduli are therefore less than the terms of a converging geometric progres
sion, so that they form a converging series; the limit of \L\, and therefore
of L, can, with indefinite increase of n, be made zero and Taylor's expansion
can be derived as before.

00

Ex. 1. Prove that the arithmetic mean of all values of z~ n 2 avzv, for points lying along

v = 0

a circle |z| = r entirely contained in the region of continuity, is an. (Rouche, Gutzmer.)
Prove also that the arithmetic mean of the squares of the moduli of all values of

00

2 avzv, for points lying along a circle z\ = r entirely contained in the region of continuity,

x = 0

is equal to the sum of the squares of the moduli of the terms of the series for a point on
the circle. (Gutzmer.)

00

Ex. 2. Prove that the function 2 anzn*,

M = 0

is finite and continuous, as well as all its derivatives, within and on the boundary of the
circle |0| = 1, provided a < 1. (Fredholm.)

* Liouville, 3dmc Ser., t. ii, (1876), pp. 291—312.

28.]

LAURENT'S EXPANSION OF A FUNCTION

47

28. Laurent's theorem is as follows*: —

A function, which is holomorphic in a part of the plane bounded by two
concentric circles with centre a and finite radii, can be expanded in the form
of a double series of integral powers, positive and negative, of z — a, the series
converging uniformly and unconditionally in the part of the plane between the
circles.

Let z be any point within the region bounded by the two circles of radii
R and R; describe two concentric circles of
radii r arid r' such that

R>r> z-a >r'> R.

Denoting by t and by s current points on the
circumference of the outer and of the inner
circles respectively, and considering the space
which lies between them and includes the point
z, we have, by § 20,

/w-oL

: — Z ZTTlJs — 2"~ Fig. 11.

a negative sign being prefixed to the second integral because the direction
indicated in the figure is the negative direction for the description of the
inner circle regarded as a portion of the boundary.

Now we have

fz — a"
t — a _ z —

t

' — a Iz — a\*
— a \t-aj

z- a.

+ I . — - +
— a.

1 -

z — a

t — a

this expansion being adopted with a view to an infinite converging series,
z — a

because

t — a

is less than unity for all points t; and hence, by § 15,

_ n\n+l

dt.

— z \t — a/

Now each of the integrals, which are the respective coefficients of powers of
z — a, is finite, because the subject of integration is everywhere finite along
the circle of finite radius, by § 15 (IV.). Let the value of

^r* % '•'-'••-

be 2iriur : the quantity ur is not necessarily equal to /'' (a) -r- r I, because no
* Comptes Rendus, t. xvii, (1843), p. 939.

48 LAURENT'S EXPANSION OF [28.

knowledge of the function or of its derivatives is given for a point within
the innermost circle of radius R'. Thus

_L f/2) dt = u0 + (z - a) u1 + (z- a)2 w2+ +(z- a)nun

2w» J t — z

1 [f (t) (z — a\n+1 -,

- z \t — a

The modulus of the last term is less than

M

where p is z-a and If is the greatest value of \f(t)\ for points along the
circle. Because p < r, this quantity diminishes to zero with indefinite in
crease of n ; and therefore the modulus of the expression

v %

becomes indefinitely small with increase of n. The quantity itself therefore
vanishes in the same limiting circumstance ; and hence

1 . [fl&dt = u0 + (z-<i)u1 + ...... +(z-a)mum+ ...... ,

2-7TI J t — Z

so that the first of the integrals is equal to a series of positive powers. This
series converges uniformly and unconditionally within the outer circle, for
the modulus of the (m + l)th term is less than

which is the (m + l)th term of a converging series*.

As in § 27, the equivalence of the integral and the series can be affirmed
only for points which lie within the outermost circle of radius R.

Again, we have

fs - a\n+1

z-a _ s-a fs - a\n (z-a)

s-z z-a \z-a

z — a

this expansion being adopted with a view to an infinite converging series,

because

s — a

z — a

is less than unity. Hence

1 [/s-a\
. If - -
2?rt J \z-aj

-n+1f(s)
J-~-

,

-ds.
z — s

Chrystal, ii, 124.

28.] A FUNCTION IN SERIES 49

The modulus of the last term is less than

M'

P

where M' is the greatest value of \f(s)\ for points along the circle of radius
r'. With indefinite increase of n, this modulus is ultimately zero ; and thus,
by an argument similar to the one which was applied to the former integral,
we have

.. .. - ..

ZTTI J s — z z — a (z — a)2 (z — a)m

where vm denotes the integral f(s — a)m~lf (s) ds taken round the circle.

As in the former case, the series is one which converges uniformly and
unconditionally; and the equivalence of the integral and the series is valid
for points z that lie without the innermost circle of radius R'.

The coefficients of the various negative powers of z — a are of the form

1 f /(*) d( 1 ^
tori] __ 1_ (s-a)'

(s - a)m
a form that suggests values of the derivatives of f (s) at the point given by

- = 0, that is, at infinity. But the outermost circle is of finite radius ;
s-a

and no knowledge of the function at infinity, lying without the circle, is
given, so that the coefficients of the negative powers may not be assumed
to be the values of the derivatives at infinity, just as, in the former case, the
coefficients ur could not be assumed to be the values of the derivatives at the
common centres of the circles.

Combining the expressions obtained for the two integrals, we have
f(z) = u0 + (z — a) u-i + (z — a)2 w2 + ...

+ (z- a)-1 Vl + (z- a)~2 va+ ....

Both parts of the double series converge uniformly and unconditionally for
all points in the region between the two circles, though not necessarily for
points on the boundary of the region. The whole series therefore converges
for all those points : and we infer the theorem as enunciated.

Conformably with the notation (§ 26, note) adopted to represent Taylor's
expansion, a function f(z) of the character required by Laurent's Theorem
can be represented in the form

the series P1 converging within the outer circle and the series P2 converging
without the inner circle ; their sum converges for the ring-space between the
circles.

F. 4

50 LAURENT'S THEOREM [29.

29. The coefficient u0 in the foregoing expansion is

-1- f £9 dt

torijt-a '

the integral being taken round the circle of radius r. We have

dt =ide

t — a
for points on the circle ; and therefore

d0

so that \u0\<!deMt<M',

J ZTT

M' being the greatest value of Mt, the modulus of f(t), for points along the
circle. If M be the greatest value of \f(z}\ for any point in the whole
region in which f(z) is defined, so that M'^.M, then we have

«o 1 < M,

that is, the modulus of the term independent of z — a in the expansion of
f(z) by Laurent's Theorem is less than the greatest value of \f(z) \ at points
in the region in which it is defined.

Again, (z-a)-mf(z) is a double series in positive and negative powers of
z-a, the term independent of z -a being um; hence, by what has just been
proved, um \ is less than p~m M, where p is z - a . But the coefficient um
does not involve z, and we can therefore choose a limit for any point z. The
lowest limit will evidently be given by taking z on the outer circle of radius
R, so that um < MR~m. Similarly for the coefficients vm ; and therefore we
have the result : —

If f(z) be expanded as by Laurent's Theorem in the form

OO 00

u0+ 2 (z-a)mum+ 2 (z-aY^Vm,

m = l m=l

then \um <MR~m, \vm <MR'm,

where M is the greatest value of \f(z) at points within the region in which

f(z) is defined, and R and R' are the radii of the outer and the inner circles

respectively.

30. The following proposition is practically a corollary from Laurent's
Theorem : —

When a function is holomorphic over all the plane which lies outside a
circle of centre a, it can be expanded in the form of a series of negative integral
powers of z — a, the series converging uniformly and unconditionally everywhere
in that part of the plane.

It can be deduced as the limiting case of Laurent's Theorem when the

30.] EXPANSION IN NEGATIVE POWERS 51

radius of the outer circle is made infinite. We then take r infinitely large,
and substitute for t by the relation

t — a = reei,

so that the first integral in the expression (a), p. 47, for/(^) is

1 f2" d0

t — a

Since the function is holomorphic over the whole of the plane which lies
outside the assigned circle, f(t} cannot be infinite at the circle of radius r
when that radius increases indefinitely. If it tend towards a (finite) limit k,
which must be uniform owing to the hypothesis as to the functional character
of f(z\ then, since the limit of (t — z)/(t — a) is unity, the preceding integral
is equal to k.

The second integral in the same expression (a), p. 47, for f(z) is un
altered by the conditions of the present proposition ; hence we have

f(z) = k + (z- a)~l vl + (z- a)-2Vz + ...,

the series converging uniformly and unconditionally without the circle,
though it does not necessarily converge on the circumference.

The series can be represented in the form

1

\z — a/
conformably with the notation of § 26.

Of the three theorems in expansion which have been obtained, Cauchy's
is the most definite, because the coefficients of the powers are explicitly
obtained as values of the function and of its derivatives at an assigned point.
In Laurent's theorem, the coefficients are not evaluated into simple expres
sions ; and in the corollary frofti Laurent's theorem the coefficients are, as is
easily proved, the values of the function and of its derivatives for infinite
values of the variable. The essentially important feature of all the theorems
is the expansibility of the function in series under assigned conditions.

31. It was proved (§21) that, when a function is holomorphic in any
region of the plane bounded by a simple curve, it has an unlimited number
of successive derivatives each of which is holomorphic in the region. Hence,
by the preceding propositions, each such derivative can be expanded in
converging series of integral powers, the series themselves being deducible
by differentiation from the series which represents the function in the region.

In particular, when the region is a finite circle of centre a, within which
f(z) and consequently all the derivatives off(z) are expansible in converging
series of positive integral powers of z — a, the coefficients of the various
powers of z — a are — save as to numerical factors — the values of the

4—2

52 DEFINITION OF DOMAIN [31.

derivatives at the centre of the circle. Hence it appears that, when a function
is holomorphic over the area of a given circle, the values of the function and all
its derivatives at any point z within the circle depend only upon the variable
of the point and upon the values of the function and its derivatives at the
centre.

32. Some of the classes of points in a plane that usually arise in
connection with uniform functions may now be considered.

(i) A point a in the plane may be such that a function of the variable
has a determinate finite value there, always independent of the path by
which the variable reaches a ; the point a, is called an ordinary point* of the
function. The function, supposed continuous in the vicinity of a, is con
tinuous at a : and it is said to behave regularly in the vicinity of an ordinary
point.

Let such an ordinary point a be at a distance d, not infinitesimal, from
the nearest of the singular points (if any) of the function ; and let a circle of
centre a and radius just less than d be drawn. The part of the z-plane lying
within this circle is calledf the domain of a ; and the function, holomorphic
within this circle, is said to behave regularly (or to be regular) in the domain
of a. From the preceding section, we infer that a function and its derivatives
can be expanded in a converging series of positive integral powers of z — a
for all points z in the domain of a, an ordinary point of the function : and
the coefficients in the series are the values of the function and its derivatives
at a.

The property possessed by the series — that it contains only positive
integral powers of z - a— at once gives a test that is both necessary and
sufficient to determine whether a point is an ordinary point. If the point a
be ordinary, the limit of (z - a) f (z} necessarily is zero when z becomes equal
to a. This necessary condition is also sufficient to ensure that the point is
an ordinary point of the function / (z), supposed to be uniform ; for, since
f(z) is holomorphic, the function (z-a)f(z) is also holomorphic and can be
expanded in a series

M0 -f wa (z — d) + w2 (? — a)2 + • • -,

converging in the domain of a. The quantity u0 is zero, being the value
of (z-a)f(z) at a and this vanishes by hypothesis; hence

(z-a)f (z) = (z — a) {MI + u2(z -a) +...},

shewing that / (z) is expressible as a series of positive integral powers of
z— a converging within the domain of a, or, in other words, that/(*) certainly
has a for an ordinary point in consequence of the condition being satisfied.

* Sometimes a regular point.

t The German title is Umgebung, the French is domaine.

32.] ESSENTIAL SINGULARITY 53

(ii) A point a in the plane may be such that a function / (z) of the
variable has a determinate infinite value there, always independent of the
path by which the variable reaches a, the function behaving regularly for

points in the vicinity of a ; then ^—\ nas a determinate zero value there, so

/ (?)

that a is an ordinary point of --r-r . The point a is called a pole (§12) or
an accidental singularity* of the function.

A test, necessary and sufficient to settle whether a point is an accidental
singularity of a function will subsequently (§ 42) be given.

(iii) A point a in the plane may be such that y (2) has not a determinate
value there, either finite or infinite, though the function is regular for all

points in the vicinity of a that are not at merely infinitesimal distances.

i 1
Thus the origin is of this nature for the functions ez, sn - .

Z

Such a point is called-f* an essential singularity of the function. No
hypothesis is postulated as to the character of the function for points
at infinitesimal distances from the essential singularity, while the relation
of the singularity to the function naturally depends upon this character at
points near it. There may thus be various kinds of essential singularities
all included under the foregoing definition ; their classification is effected
through the consideration of the character of the function at points in their
immediate vicinity. (See § 88.)

One sufficient test of discrimination between an accidental singularity
and an essential singularity is furnished by the determinateness of the value
at the point. If the reciprocal of the function have the point for an ordinary
point, the point is an accidental singularity — it is, indeed, a zero for the
reciprocal. But when the point is an essential singularity, the value of the
reciprocal of the function is not determinate there ; and then the reciprocal,
as well as the function, has the point for an essential singularity.

33. It may be remarked at once that there must be at least one
infinite value among the values which a function can assume at an essential
singularity. For if/ (z) cannot be infinite at a, then the limit of (z — a)f (z)
is zero when z = a, no matter what the non-infinite values of f (z) may be,
that is, the limit is a determinate zero. The function (z — a)f(z) is regular
in the vicinity of a : hence by the foregoing test for an ordinary point,
the point a is ordinary and the value of the uniform function f(z) is

* Weierstrass, Abh. aus der Functionenlehre, p. 2, to whom the name is due, calls it ausser-
wesentliche singuldre Stelle ; the term non-essential is suggested by Mr Cathcart, Harnack, p. 148.
t Weierstrass, I.e., calls it wesentliche singulare Stelle.

54 CONTINUATIONS OF A FUNCTION [33.

determinate, contrary to hypothesis. Hence the function must have at least
one infinite value at an essential singularity.

Further, a uniform function must be capable of assuming any value C at
an essential singularity. For an essential singularity of / (z) is also an

essential singularity of / (z) — G and therefore also of .. \_n • The last

function must have at least one infinite value among the values that it can
assume at the point ; and, for this infinite value, we have / (z) — C at the
point, so that/(f) assumes the assigned value C at the essential singularity*.

34. Let f(z) denote the function represented by a series of powers
Pj (z — a), the circle of convergence of which is the domain of the ordinary
point a of the function. The region over which the function / (z) is holo-
morphic may extend beyond the domain of a, although the circumference
bounding that domain is the greatest of centre a that can be drawn within
the region. The region evidently cannot extend beyond the domain of a in
all directions.

Take an ordinary point b in the domain of a. The value at b of the
function /(V) is given by the series Pj (b — a), and the values at b of all its
derivatives are given by the derived series. All these series converge within
the domain of a and they are therefore finite at b ; and their expressions
involve the values at a of the function and its derivatives.

Let the domain of b be formed. The domain of b may be included in
that of a, and then its bounding circle will touch the bounding circle of the
domain of a internally. If the domain of b be not entirely included in that
of a, part of it will lie outside the domain of a ; but it cannot include the
whole of the domain of a unless its bounding circumference touch that of the
domain of a externally, for otherwise it would extend beyond a in all
directions, a result inconsistent with the construction of the domain of a.
Hence there must be points excluded from the domain of a which are also
excluded from the domain of b.

For all points z in the domain of b, the function can be represented by a
series, say P2 (2 — b), the coefficients of which are the values at b of the
function and its derivatives. Since these values are partially dependent
upon the corresponding values at a, the series representing the function may
be denoted by P2 (z — b, a).

At a point z in the domain of b lying also in the domain of a, the two
series Pl (z — a) and P2 (z — b, a) must furnish the same value for the
function / (V) ; and therefore no new value is derived from the new series P2

* Weierstrass, I.e., pp. 50—52; Durege, Elemente der Theorie der Funktionen, p. 119; Holder,
Math. Ann., t. xx, (1882), pp. 138 — 143 ; Picard, " Memoire sur les fonctions entieres," Annahs de
VEcole Norm. Sup., 2me Ser., t. ix, (1880), pp. 145 — 166, which, in this regard, should be consulted
in connection with the developments in Chapter V. See also § 62.

34.] OVER ITS REGION OF CONTINUITY 55

which cannot be derived from the old series Pj. For all such points the new
series is of no advantage ; and hence, if the domain of b be included in that
of a, the construction of the series P2 (z — b, a) is superfluous. Hence in
choosing the ordinary point b in the domain of a we choose a point, if
possible, that will not have its domain included in that of a.

At a point z in the domain of b, which does not lie in the domain of a,
the series P2 (z — b, a) gives a value for f(z) which cannot be given by
Pl (z — a). The new series P2 then gives an additional representation of the
function ; it is called* a continuation of the series which represents the function
in the domain of a. The derivatives of P2 give the values of f(z) for points
in the domain of b.

It thus appears that, if the whole of the domain of b be not included in
that of a, the function can, by the series which is valid over the whole
of the new domain, be continued into that part of the new domain excluded
from the domain of a.

Now take a point c within the region occupied by the combined domains
of a and b ; and construct the domain of c. In the new domain, the function
can be represented by a new series, say P3(z — c), or, since the coefficients
(being the values at c of the function and of its derivatives) involve the
values at a and possibly also the values at b of the function and of its
derivatives, the series representing the function may be denoted by
Pz(z — c, a, b). Unless the domain of c include points, which are not
included in the combined domains of a and b, the series P3 does not give
a value of the function which cannot be given by Pj or P2: we therefore
choose c, if possible, so that its domain will include points not included in
the earlier domains. At such points z in the domain of c as are excluded
from the combined domains of a and 6, the series P3 (z — c, a, b) gives a value
for f(z) which cannot be derived from P1 or P2 ; and thus the new series
is a continuation of the earlier series.

Proceeding in this manner by taking successive points and constructing
their domains, we can reach all parts of the plane connected with one
another where the function preserves its holomorphic character; their
combined aggregate is called -f the region of continuity of the function.
With each domain, constructed so as to include some portion of the region of
continuity not included in the earlier domains, a series is associated, which is
a continuation of the earlier series and, as such, gives a value of the function
not deducible from those earlier series ; and all the associated series are
ultimately derived from the first.

* Biermann, Theorie der analytischen Functional, p. 170, which may be consulted in
connection with the whole of § 34; the German word is Fortsetzung.
t Weierstrass, I.e., p. 1.

56 DEFINITION OF ANALYTIC FUNCTION [34.

Each of the continuations is called an Element of the function. The
aggregate of all the distinct elements is called a monogenic analytic function :
it is evidently the complete analytical expression of the function in its region
of continuity.

Let z be any point in the region of continuity, not necessarily in the
circle of convergence of the initial element of the function; a value of the
function at z can be obtained through the continuations of that initial
element. In the formation of each new domain (and therefore of each new
element) a certain amount of arbitrary choice is possible ; and there may,
moreover, be different sets of domains which, taken together in a set, each
lead to z from the initial point. When the analytic function is uniform, as
before defined (§ 12), the same value at z for the function is obtained,
whatever be the set of domains. If there be two sets of elements, differently
obtained, which give at z different values for the function, then the ana
lytic function is multiform, as before defined (§ 12) ; but not every change
in a set of elements leads to a change in the value at z of a multiform
function, and the analytic function is uniform within such a region of the
plane as admits only equivalent changes of elements.

The whole process is reversible when the function is uniform. We can
pass back from any point to any earlier point by the use, if necessary, of
intermediate points. Thus, if the point a in the foregoing explanation
be not included in the domain of b (there supposed to contribute a continu
ation of the first series), an intermediate point on a line, drawn in the
region of continuity so as to join a and b, would be taken ; and so on,
until a domain is formed which does include a. The continuation, associated
with this domain, must give at a the proper value for the function and its
derivatives, and therefore for the domain of a the original series Pl(z — a)
will be obtained, that is, Pj (z — a) can be deduced from P2 (z — b, a) the
series in the domain of b. This result is general, so that any one of the
continuations of a uniform function, represented by a power-series, can be
derived from any other; and therefore the expression of such a function in
its region of continuity is potentially given by one element, for all the
distinct elements can be derived from any one element.

35. It has been assumed that the property, characteristic of some of the
functions adduced as examples, of possessing either accidental or essential
singularities, is characteristic of all functions ; it will be proved (§ 40) to hold
for every uniform function which is not a mere constant.

The singularities limit the region of continuity ; for each of the separate
domains is, from its construction, limited by the nearest singularity, and the
combined aggregate of the domains constitutes the region of continuity when

35.]

SCHWARZ S CONTINUATION

57

they form a continuous space*. Hence the complete boundary of the region
of continuity is the aggregate of the singularities of the function-}-.

It may happen that a function has no singularity except at infinity ; the
region of continuity then extends over the whole finite part of the plane but
it does not include the point at infinity.

It follows from the foregoing explanations that, in order to know a
uniform analytic function, it is necessary to know some element of the
function, which has been shewn to be potentially sufficient for the derivation
of the full expression of the function and for the construction of its region of
continuity.

36. The method of continuation of a function, which has just been
described, is quite general ; there is one particular continuation, which is
important in investigations on conformal representations. It is contained in
the following proposition, due to SchwarzJ : —

If an analytic function w of z be defined only for a region 8' in the
positive half of the z-plane and if continuous real values of w correspond to
continuous real values of z, then w can be continued across the axis of real
quantities.

Consider a region 8", symmetrical with S' relative to the axis of real
quantities (fig. 12). Then a function is
defined for the region S" by associating
a value w0, the conjugate of w, with z0,
the conjugate of z.

Let the two regions be combined along
the portion of the axis of ac which is their
common boundary ; they then form a
single region S' + S".

Consider the integrals

Fig. 12.

1 [ w j A ! [ wo
o — • I — i-dz and ^ — -. / —
fcp/fjr-f 2w»./,r«t-

taken round the boundaries of 8' and of 8" respectively. Since w is

* Cases occur in which the region of continuity of a function is composed of isolated spaces,
each continuous in itself, but not continuous into one another. The consideration of such cases
will be dealt with briefly hereafter, and they are assumed excluded for the present : meanwhile,
it is sufficient to note that each continuous space could be derived from an element belonging to
some domain of that space and that a new element would be needed for a new space.

t See Weierstrass, I.e., pp. 1—3 ; Mittag-Leffler, " Sur la representation analytique des fonctions
monogenes uniformes d'une variable independante," Acta Math., t. iv, (1884), pp. 1 et seq.,
especially pp. 1 — 8.

£ Crelle, t. Ixx, (1869), pp. 106, 107, and Ges. Math. Abh., t. ii, pp. 66—68. See also Darboux,
Theorie generate des surfaces, t. i, § 130.

58 SCHWARZ'S CONTINUATION [36.

continuous over the whole area of 8' as well as along its boundary and
likewise w0 relative to 8", it follows that, if the point f be in 8', the value of
the first integral is w (f ) and that of the second is zero ; while, if £ lie in 8",
the value of the first integral is zero and that of the second is w0 (£). Hence
the sum of the two integrals represents a unique function of a point in either
8' or 8". But the value of the first integral is

M wdz J^ [B w Q) dap

I ("' ~~ (f • C\ ' I V>

J ji 2— £ ZTriJ A x — L,

the first being taken along the curve EC. A and the second along the axis
AxB ; and the value of the second integral is

1 CAw0(x)dx 1_ f * W0dz0
2-Tri J B x — £ ZTTI J A *o — £ '
the first being taken along the axis Ex A and the second along the curve

ADB. But

w0 (ac) = w (x),

because conjugate values w and w0 correspond to conjugate values of the
argument by definition of W0 and because w (and therefore also w0) is real
and continuous when the argument is real and continuous. Hence when the
sum of the four integrals is taken, the two integrals corresponding to the
two descriptions of the axis of x cancel and we have as the sum

wdz 1

A

and this sum represents a unique function of a point in 8' + 8". These two
integrals, taken together, are

_L [w'dz
2Tn]z-t'

taken round the whole contour of 8' + 8", where w' is equal to w (f) in the
positive half of the plane and to w0 (^) in the negative half.

For all points £ in the whole region 8' + 8", this integral represents a
single uniform, finite, continuous function of f; its value is w (£) in the
positive half of the plane and is w0 (f) in the negative half; and therefore
w0 (£) is the continuation into the negative half of the plane of the function,
which is defined by w (£) for the positive half.

For a point c on the axis of x, we have

w (z) -w(c) = A(z-c) + B(z-cy>+C(z-cY + ...;

and all the coefficients A, B, C,... are real. If, in addition, w be such a
function of z that the inverse functional relation makes z a uniform
analytic function of w, it is easy to see that A must not vanish, so that the
functional relation may be expressed in the form

w(z)—w (c) = (z-c}P(z- c),
where P (z — c) does not vanish when z = c.

CHAPTER IV.

GENERAL PROPERTIES OF UNIFORM FUNCTIONS, PARTICULARLY OF THOSE
WITHOUT ESSENTIAL SINGULARITIES.

37. IN the derivation of the general properties of functions, which will be
deduced in the present and the next three chapters from the results already
obtained, it is to be supposed, in the absence of any express statement to
other effect, that the functions are uniform, monogenic and, except at either
accidental or essential singularities, continuous*.

THEOREM I. A function, which is constant throughout any region of the
plane not infinitesimal in area, or which is constant along any line not infini
tesimal in length, is constant throughout its region of continuity.

For the first part of the theorem, we take any point a in the region of the
plane where the function is constant, and we draw a circle of centre a and
of any radius, provided only that the circle remains within the region of
continuity of the function. At any point z within this circle we have

/<*) =/(a) + (z - a)f (a) + (-, ~^ f" (a) + . . ,

a converging series the coefficients of which are the values of the function
and its derivatives at a. But

/X«) = Limit of ^±MZ/^), :. V, ' :•

which is zero because f(a + Ba) is the same constant as f(a) : so that the
first derivative is zero at a. Similarly, all the derivatives can be shewn to
be zero at a ; hence the above series after its first term is evanescent,
and we have

/(*)-/<«),

that is, the function preserves its constant value throughout its region of
continuity.

The second result follows in the same way, -when once the derivatives are
proved zero. Since the function is monogenic, the value of the first and

* It will be assumed, as in § 35 (note, p. 57), that the region of continuity consists of a single
space ; functions, with regions of continuity consisting of a number of separated spaces, will be
discussed in Chap. VII.

60 ZEROS OF A [37.

of each of the successive derivatives will be obtained, if we make the
differential element of the independent variable vanish along the line.

Now, if a be a point on the line and a + 8a a consecutive point, we have
f(a + So) = f(a) ; hence /' (a) is zero. Similarly the first derivative at any
other point on the line is zero. Therefore we have /' (a + So) =f (a), for
each has just been proved to be zero : hence /" (a) is zero ; and similarly the
value of the second derivative at any other point on the line is zero. So on
for all the derivatives : the value of each of them at a is zero.

Using the same expansion as before and inserting again the zero values
of all the derivatives at a, we find that

/(*)=/(«),

so that under the assigned condition the function preserves its constant value
throughout its region of continuity.

It should be noted that, if in the first case the area be so infinitesimally
small and in the second the line be so infinitesimally short that consecutive
points cannot be taken, then the values at a of the derivatives cannot be
proved to be zero and the theorem cannot then be inferred.

COROLLARY I. If two functions have the same value over any area of
their common region of continuity which is not infinitesimally small or along
any line in that region which is not infinitesimally short, then they have the
same values at all points in their common region of continuity.

This is at once evident : for their difference is zero over that area or along
that line and therefore, by the preceding theorem, their difference has a
constant zero value, that is, the functions have the same values, everywhere
in their common region of continuity.

But two functions can have the same values at a succession of isolated
points, without having the same values everywhere in their common region
of continuity ; in such a case the theorem does not apply, the reason being
that the fundamental condition of equality over a continuous area or along
a continuous line is not satisfied.

COROLLARY II. A function cannot be zero over any continuous area of its
region of continuity which is not infinitesimal or along any line in that region
which is not infinitesimally short without being zero everywhere in its region of
continuity.

This corollary is deduced in the same manner as that which precedes.

If, then, there be a function which is evidently not zero everywhere, we
conclude that its zeros are isolated points though such points may be multiple
zeros.

Further, in any finite area of the region of continuity of a function that is
subject to variation, there can be at most only a finite number of its zeros, when

37.] UNIFORM FUNCTION 61

no point of the boundary of the area is infinitesimally near an essential
singularity. For if there were an infinite number of such points in any
such region, there must be a cluster in at least one area or a succession
along at least one line, infinite in number and so close as to constitute a
continuous area or a continuous line where the function is everywhere zero.
This would require that the function should be zero everywhere in its region
of continuity, a condition excluded by the hypothesis.

And it immediately follows that the points (other than those infini
tesimally near an essential singularity) in a region of continuity, at which a
function assumes any the same value, are isolated points ; and that only a
finite number of such points occur in any finite area.

38. THEOREM II. The multiplicity m of any zero a of a function is
finite provided the zero be an ordinary point of the function, which is not zero
throughout its region of continuity; and the function can be expressed in the

where <f> (z) is holomorphic in the vicinity of a, and a is not a zero of <£ (z).

Let f(z) denote the function ; since a is a zero, we have f(a) = 0.
Suppose that /'(a), f" (a), ...... vanish: in the succession of the derivatives

of f, one of finite order must be reached which does not have a zero value.
Otherwise, if all vanish, then the function and all its derivatives vanish at a;
the expansion of f(z) in powers of z — a leads to zero as the value of f (z\
that is, the function is everywhere zero in the region of continuity, if all the
derivatives vanish at a.

Let, then, the wth derivative be the first in the natural succession which
does not vanish at a, so that m is finite. Using Cauchy's expansion, we have

(? — n\tm) ( ~ _ n\(m+\)

f(z) = (Z a /« (a) + S£Za_/F* (a) + . . .

J m ! J (m + 1) ! J

= (z-ay*$(z\

where <£ (z) is a function that does not vanish with a and, being the quotient
of a converging series by a monomial factor, is holomorphic in the immediate
vicinity of a.

COROLLARY I. If infinity be a zero of a function of multiplicity m and
at the same time be an ordinary point of the function, then the function can be

expressed in the form z~m $ f-J ,

where </>(-) is a function that is continuous and non-evanescent for infinitely

large values of z.

The result can be derived from the expansion in § 30 in the same way as
the foregoing theorem from Cauchy's expansion.

62 ZEROS OF A [38.

COROLLARY II. The number of zeros of a function, account being taken of
their multiplicity, which occur within a finite area of the region of continuity
of the function, is finite, when no point of the boundary of the area is infinitesi-
mally near an essential singularity.

By Corollary II. of § 37, the number of distinct zeros in the limited area
is finite, and, by the foregoing theorem, the multiplicity of each is finite ;
hence, when account is taken of their respective multiplicities, the total
number of zeros is still finite.

The result is, of course, a known result for an algebraical polynomial ; but
the functions in the enunciation are not restricted to be of the type of
algebraical polynomials.

Note. It is important to notice, both for the Theorem and for Corollary I,
that the zero is an ordinary point of the function under consideration ; the
implication therefore is that the zero is a definite zero and that in the
immediate vicinity of the point the function can be represented in the form

P(z — a) or P [-] , the function P(a — a) or P (— ) being .always a definite

\<6 / \ /

zero.

Instances do occur for which this condition is not satisfied. The point
may not be an ordinary point, and the zero value may be an indeterminate
zero ; or zero may be only one of a set of distinct values though everywhere
in the vicinity the function is regular. Thus the analysis of § 13 shews that

z=a is a point where the function sn - - has any number of zero values and

Z CL

any number of infinite values, and there is no indication that there are not
also other values at the point. In such a case the preceding proposition does
not apply ; there may be no limit to the order of multiplicity of the zero, and
we certainly cannot infer that any finite integer m can be obtained such that

(z - a)~m <j> (z)

is finite at the point. Such a point is (§ 32) an essential singularity of the
function.

39. THEOREM III. A multiple zero of a function is a zero of its
derivative ; and the multiplicity for the derivative is less or is greater by
unity according as the zero is not or is at infinity.

If a be a point in the finite part of the plane which is a zero of f(z)
of multiplicity n, we have

/(f)-(*T.a)» + («X

and therefore /' (z) = (z - a)n~l [n$ (z} + (z-a) $ (z)}.
The coefficient of (z — a)n~l is holomorphic in the immediate vicinity of a and
does not vanish for a ; hence a is a zero for /' (z) of decreased multiplicity

39.] UNIFORM FUNCTION

If z = oo be a zero off(z) of multiplicity r, then

where <£ (-) is holomorphic for very large values of z and does not vanish at

\z /

infinity. Therefore

The coefficient of ^~r~1 is holomorphic for very large values of z, and does
not vanish at infinity ; hence z=<x> is a zero off (z) of increased multiplicity
r + l.

Corollary I. If a function be finite at infinity, then z = oo is a zero of the
first derivative of multiplicity at least two.

Corollary II. If a be a finite zero off(z) of multiplicity n, we have
f(z)= n #(z)
f(z) ir-** fW

Now a is not a zero of <J> (z) ; and therefore ^4^r is finite, continuous, uniform

9W

and monogenic in the immediate vicinity of a. Hence, taking the integral
of both members of the equation round a circle of centre a and of radius
so small as to include no infinity and no zero, other than a, of / (z) _ and
therefore no zero of $(z) — we have, by § 17 and Ex. 2, § 25,

~jT/ \ ^"^ ~ ^-

/(*)

40. THEOREM IV. A function must have an infinite value for some finite
or infinite value of the variable.

If M be a finite maximum value of the modulus for points in the plane,
then (§ 22) we have

where r is the radius of an arbitrary circle of centre a, provided the whole of
the circle is in the region of continuity of the function. But as the function
is uniform, monogenic, finite and continuous everywhere, this radius can be
increased indefinitely ; when this increase takes place, the limit of

is zero and therefore /<»> (a) vanishes. This is true for all the indices 1,2,...
of the derivatives.

64 INFINITIES OF A [40.

Now the function can be represented at any point z in the vicinity of a
by the series

which degenerates, under the present hypothesis, to /(a), so that the function
is everywhere constant. Hence, if a function has not an infinity somewhere
in the plane, it must be a constant.

The given function is not a constant; and therefore there is no finite
limit to the maximum value of its modulus, that is, the function acquires
an infinite value somewhere in the plane.

COROLLARY I. A function must have a zero value for some finite or
infinite value of the variable.

For the reciprocal of a uniform monogenic analytic function is itself a
uniform monogenic analytic function ; and the foregoing proposition shews
that this reciprocal must have an infinite value for some value of the
variable, which therefore is a zero of the function.

COROLLARY II. A function must assume any assigned value at least once.

COROLLARY III. Every function which is not a mere constant must have
at least one singularity, either accidental or essential. For it must have
an infinite value : if this be a determinate infinity, the point is an accidental
singularity (§ 32) ; if it be an infinity among a set of values at the point, the
point is an essential singularity (§§ 32, 33).

41. Among the infinities of a function, the simplest class is that con
stituted by its accidental singularities, already defined (§ 32) by the property
that, in the immediate vicinity of such a point, the reciprocal of the function
is regular, the point being an ordinary (zero) point for that reciprocal.

THEOREM V. A function, which has a point cfor an accidental singularity,
can be expressed in the foi*m

(z - c}~n (f> (z),

where n is a finite positive integer and <f> (z) is a continuous function in the
vicinity of c.

Since c is an accidental singularity of the function f(z}, the function ^y-r

/ (z)

is regular in the vicinity of c and is zero there (§ 32). Hence, by § 38, there
is a finite limit to the multiplicity of the zero, say n (which is a positive
integer), and we have

where ^ (z) is uniform, monogenic and continuous in the vicinity of c and is
not zero there. The reciprocal of ^ (z), say <f> (z), is also uniform, monogenic

41.] UNIFORM FUNCTION 65

and continuous in the vicinity of c, which is an ordinary point for (f> (z) ;
hence we have

f(z} = (Z-c)-^(z\

which proves the theorem.

The finite positive integer n measures the multiplicity of the accidental
singularity at c, which is sometimes said to be of multiplicity n or of
order n.

Another analytical expression for f(z) can be derived from that which
has just been obtained. Since c is an ordinary point for <f> (z) and not a zero,
this function can be expanded in a series of ascending, positive, integral
powers of z — c, converging in the vicinity of c, in the form

£(*) = P(*-c)

= uQ + ul(z-c} + ... + un^(z-c)n-l+un(z-c)n+...

= u0 + u,(z - c) + ... + un_^(z - c)71-1 + (z- c)nQ(z-c),
where Q(z — c), a series of positive, integral, powers of z — c converging in the
vicinity of c, is a monogenic analytic function of z. Hence we have

^ = ^» + (7^+ - +,~; + «('-')>

the indicated expression for f(z), valid in the immediate vicinity of c, where
Q (z — c) is uniform, finite, continuous and monogenic.

COROLLARY. A function, which has z= oo for an accidental singularity of
multiplicity n, can be expressed in the form

_

where </>(-) is a continuous function for very large values of \z , and is not

\zj

zero when z = oo . It can also be expressed in the form

1 + ... + an^ z + Q (-} ,
\zj

where Q ( - j is uniform, finite, continuous and monogenic for very large values

f\»\.

The derivation of the form of the function in the vicinity of an accidental
singularity has been made to depend upon the form of the reciprocal of the
function. Whatever be the (finite) order of that point as a zero of the
reciprocal, it is assumed that other zeros of the reciprocal are not at merely
infinitesimal distances from the point, that is, that other infinities of the
function are not at merely infinitesimal distances from the point.

Hence the accidental singularities of a function are isolated points ; and
there is only a finite number of them in any limited portion of the plane.
F. 5

66 INFINITIES OF A [42.

42. We can deduce a criterion which determines whether a given singu
larity of a function /(f) is accidental or essential.

When the point is in the finite part of the plane, say at c, and a finite
positive integer n can be found such that

is not infinite at c, then c is an accidental singularity.

When the point is at infinity and a finite positive integer n can be found
such that

is not infinite when z = oc , then z = oo is an accidental singularity.

If one of these conditions be not satisfied, the singularity at the point is
essential. But it must not be assumed that the failure of the limitation to
finiteness in the multiplicity of the accidental singularity is the only source
or the complete cause of essential singularity.

Since the association of a single factor with the function is effective in
preventing an infinite value at the point when one of the conditions is
satisfied, it is justifiable to regard the discontinuity of the function at
the point as not essential and to call the singularity either non-essential
or accidental (§ 82).

43. THEOREM VI. The poles of a function, that lie in the finite part
of the plane, are all the poles (of increased multiplicity) of the derivatives of
the function that lie in the finite part of the plane.

Let c be a pole of the function f(z) of multiplicity p : then, for any point
z in the vicinity of c,

where </> (z) is holomorphic in the vicinity of c, and does not vanish for z = c.
Then we have

f'(2) = (z~ c)~p $' (z) ~ P (2 ~ c) p 1 $ W
= (z-c)-P-*{(z-c)<j>'(z)-p<}>(z)}

where % (z) is holomorphic in the vicinity of c, and does not vanish for z = c.

Hence c is a pole of/' (z) of multiplicity ^9 + 1. Similarly it can be shewn
to be a pole of /(r) (z) of multiplicity p + r.

This proves that all the poles of f(z) in the finite part of the plane are
poles of its derivatives. It remains to prove that a derivative cannot have
a pole which the original function does not also possess.

Let a be a pole off'(z) of multiplicity m : then, in the vicinity of a,f'(z)
can be expressed in the form

43.] UNIFORM FUNCTION £7

where ^ (z) is holomorphic in the vicinity of a and does not vanish for z = a
Thus

and therefore f (*) = - . + j_ ,

y v ' JlV^ •<*-«)*"*

so that, integrating, we have

f(z}= *(«) _*>)

m 0 - a)™-1 (m - 1) 0 - a)™-2
that is, a is a pole of/0).

An apparent exception occurs in the case when m is unity: for then
we have

the integral of which leads to

f(z} = ^ (a) log (z - a) + . . . ,

so that/0) is no longer uniform, contrary to hypothesis. Hence a derivative
cannot have a simple pole in the finite part of the plane ; and so the exception
is excluded.

The theorem is thus proved.

COROLLARY I. The rth derivative of a function cannot have a pole in the
finite part of the plane of multiplicity less than r + 1.

COROLLARY II. If c be a pole of f (z) of any order of multiplicity ^ and
if f(r] (z) be expressed in the form

, _ Oi__

» / _. _\., _!_*• _ 1 I ••••••*

(Z - CY+T (Z-

there are no terms in this expression with the indices - 1, - 2, ...... , - r.

COROLLARY III. If c be a pole of/ (z) of multiplicity p, we have

=
f(z) z-c~* 4>(z)'

where $ (z) is a holomorphic function that does not vanish for z = c, so that

<£' 0) •

-T-/JN is a holomorphic function in the vicinity of c. Taking the integral of

f'(z)

-j-j~\ round a circle, with c for centre, with radius so small as to exclude all

other poles or zeros of the function f (z), we have

5—2

(}8 INFINITIES OF A [43.

COROLLARY IV. If a simple closed curve include a number N of zeros of
a uniform function f (z) and a number P of its poles, in both of which
numbers account is taken of possible multiplicity, and if the curve contain
no essential singularity of the function, then

the integral being taken round the curve.

f (z)
The only infinities of the function ' ^i within the curve are the zeros

j(z)

and the poles of / (z). Round each of these draw a circle of radius so small
as to include it but no other infinity ; then, by Cor. II. § 18, the integral
round the closed curve is the sum of the values when taken round these
circles. By the Corollary II. § 39 and by the preceding Corollary III., the
sum of these values is

= 2w — %>

= N-P.

It is easy to infer the known theorem that the number of roots of an
algebraical polynomial of order n is n, as well as the further result that
2^ (N - P) is the variation of the argument of / (z) as z describes the
closed curve in a positive sense.

Ex. Prove that, if F(z) be holomorphic over an area, of simple contour, which con
tains roots «!, «2,... of multiplicity m» m2,... and poles cx, c2)... of multiplicity p^ p2J...
respectively of a function f(z) which has no other singularities within the contour, then

the integral being taken round the contour.

In particular, if the contour contains a single simple root a and no singularity, then that
root is given by

the integral being taken as before. (Laurent.)

44. THEOREM VII. If infinity be a pole of f (z), it is also a pole of
f (z) only when it is a multiple pole of f (z).

Let the multiplicity of the pole for f (z) be ?i; then for very large values
of z we have

/(*)-*•*£),

where <j> is holomorphic for very large values of z and does not vanish at
infinity ; hence

A«)**" •*-*'•

44.] UNIFORM FUNCTION 69

The coefficient of zn~* is holomorphic for very large values of z and does not
vanish at infinity ; hence infinity is a pole of/' (z} of multiplicity n — 1.

If n be unity, so that infinity is a simple pole of / (z), then it is not a
pole of/' (2); the derivative is then finite at infinity.

45. THEOREM VIII. A function, which has no singularity in a finite
part of the plane, and has z = oo for a pole, is an algebraical polynomial.

Let n, necessarily a finite integer, be the order of multiplicity of the pole
at infinity : then the function / (z) can be expressed in the form

1 + ...... +an^z + Q - ,

\zJ

where Q (- J is a holomorphic function for very large values of z, and is finite
(or zero) when z is infinite.

Now the first n terms of the series constitute a function which has no
singularities in the finite part of the plane : and / (z) has no singularities

in that part of the plane. Hence Q ( - J has no singularities in the finite part

of the plane : it is finite for infinite values of z. It thus can never have an
infinite value: and it is therefore merely a constant, say an. Then

/ (z) = a,zn + a^-1 + ...... + an^z + an,

a polynomial of degree equal to the multiplicity of the pole at infinity,
supposed to be the only pole of the function.

46. The above result may be obtained in the following manner.

Since z = GO is a pole of multiplicity n, the limit of z~nf (z} is not infinite
when z = oo .

Now in any finite part of the plane the function is everywhere finite, so
that we can use the expansion

where £ = *'""> dt

''+l t-z'

the integral being taken round a circle of any radius r enclosing the point z
and having its centre at the origin. As the subject of integration is finite
everywhere along the circumference, we have, by Darboux's expression in
(IV.) S 14,

T»i T _ z

where r is some point on the circumference and X is a quantity of modulus
not greater than unity.

70 TRANSCENDENTAL AND [46.

Let T = reia- ; then

X . fM

"• 71-4-1 °flii »/ \ /

'?* rn

r

f(T\
By definition, the limit of n as T (and therefore r) becomes infinitely

(£ -\—1
1 -- e~ai } is unity.
r J

Since \ is not greater than unity, the limit of \jr in the same case is zero ;
hence with indefinite increase of r, the limit of R is zero and so

shewing as before that/(^) is an algebraical polynomial.

47. As the quantity n is necessarily a positive integer*, there are two
distinct classes of functions discriminated by the magnitude of n.

The first (and the simpler) is that for which n has a finite value. The
polynomial then contains only a finite number of terms, each with a positive
integral index ; and the function is then a rational, integral, algebraical
polynomial of degree n.

The second (and the more extensive, as significant functions) is that
for which n has an infinite value. The point z = oo is not a pole, for then
the function does not satisfy the test of § 42 : it is an essential singularity
of the function, which is expansible in an infinite converging series
of positive integral powers. To functions of this class the general term
transcendental is applied.

The number of zeros of a function of the former class is known : it is
equal to the degree of the function. It has been proved that the zeros of a
transcendental function are isolated points, occurring necessarily in finite
number in any finite part of the region of continuity of the function, no
point on the boundary of the part being infinitesimally near an essential
singularity ; but no test has been assigned for the determination of the total
number of zeros of a function in an infinite part of the region of con
tinuity.

Again, when the zeros of a polynomial are given, a product-expression can
at once be obtained that will represent its analytical value. Also we know
that, if a be a zero of any uniform analytic function of multiplicity n, the
function can be represented in the vicinity of a by the expression

(x-a}n<t>(z\

where <£ (z) is holomorphic in the vicinity of a. The other zeros of the
function are zeros of <f> (z) ; this process of modification in the expression

* It is unnecessary to consider the zero value of n, for the function is then a polynomial of
order zero, that is, it is a constant.

47.] ALGEBRAICAL UNIFORM FUNCTIONS 71

can be continued for successive zeros so long as the number of zeros taken
account of is limited. But when the number of zeros is unlimited, then the
inferred product-expression for the original function is not necessarily a
converging product; and thus the question of the formal factorisation of a
transcendental function arises.

48. THEOREM IX. A function, all the singularities of which are accid
ental, is a rational, algebraical, meromorphic function.

Since all the singularities are accidental, each must be of finite
multiplicity ; and therefore infinity, if an accidental singularity, is of finite
multiplicity. All the other poles are in the finite part of the plane ; they
are isolated points and therefore only finite in number, so that the total
number of distinct poles is finite and each is of finite order. Let them be
«!, a2, ...... , a^ of orders m1} m2, ...... , m^ respectively : let m be the order of

the pole at infinity: and let the poles be arranged in the sequence of
decreasing moduli such that [aj > aF_! > ...... >|&i|-

Then, since infinity is a pole of order m, we have

/ 0) = amzm + a^z™-1 + ...... + a^z + /„ <»,

where /„ (z) is not infinite for infinite values of z. Now the polynomial

m

Sttj^ is not infinite for any finite value of z ; hence f0 (z) is infinite for all

i = l

the finite infinities of f (z) and in the same way, that is, the function f0(z)
has «!, ...... , a^ for its poles and it has no other singularities.

Again, since «M is a finite pole of multiplicity WM, we have

where fi(z) is not infinite for z = all and, as f0(z) is not infinite for z=<x> ,
evidently f^ (z) is not infinite for z = oo . Hence the singularities of f^ (z) are
merely the poles a1} ...... , aF_i ; and these are all its singularities.

Proceeding in this manner for the singularities in succession, we ultimately
reach a function f^ (z) which has only one pole a^ and no other singularity,
so that

k k

where g (z) is not infinite for z = a^ But the function f^(z) is infinite only
for 2 = 0,!, and therefore g (2) has no infinity. Hence g (z} is only a constant,
say k0 : thus

9 (*} = ^o-

Combining all these results we have a, finite number of series to add together:
and the result is that

72 UNIFORM [48.

where g1 (z) is the series k0 + a-^z + + amzm, and \ I is the sum of the

finite number of fractions. Evidently gs (z) is the product

{z — Oi)m> (z — a2)ma (z — aM)mfx ;

and g» (z) is at most of degree

If F (z} denote g1 (z} g3 (z) + g^ (z), the form of / (z) is

</.(*)'

that is, f (z) is a rational, algebraical, meromorphic function.

It is evident that, when the function is thus expressed as an algebraical
fraction, the degree of F (z) is the sum of the multiplicities of all the poles
when infinity is a pole.

COROLLARY I. A function, all the singularities of which are accidental,
has as many zeros as it has accidental singularities in the plane.

If z = oo be a pole, then it follows that, because f(z) can be expressed
in the form

it has as many zeros as F(z), unless one such should be also a zero of g^(z).
But the zeros of g3(z) are known, and no one of them is a zero of F(z), on
account of the form of f(z} when it is expressed in partial fractions. Hence
the number of zeros off(z) is equal to the degree of F(z}, that is, it is equal
to the number of poles off(z}.

If 2=00 be not a pole, two cases are possible; (i) the function f (z) may be
finite for z = oo , or (ii) it may be zero for z = oo . In the former case, the
number of zeros is, as before, equal to the degree of F (z), that is, it is equal
to the number of infinities.

In the latter case, if the degree of the numerator F (z) be K less than
that of the denominator gs (z), then z = oo is a zero of multiplicity K ; and it
follows that the number of zeros is equal to the degree of the numerator
together with K, so that their number is the same as the number of accidental
singularities.

COROLLARY II. At the beginning of the proof of the theorem of the
present section, it is proved that a function, all the singularities of which are
accidental, has only a finite number of such singularities.

Hence, by the preceding Corollary, such a function can have only a finite
number of zeros.

If, therefore, the number of zeros of a function be infinite, the function
must have at least one essential singularity.

48.] ALGEBRAICAL FUNCTIONS 73

COROLLARY III. When a uniform analytic function has no essential
singularity, if the (finite) number of its poles, say clv.., cm, be m, no one of
them being at z = oo , and if the number of its zeros, say aly..., am, be also m,
no one of them being at z = oo , then the function is

„ n

* a

r=l \Z - CT

except possibly as to a constant factor.

When z = oo is a zero of order n, so that the function has m — n zeros, say
«i, a2,..., in the finite part of the plane, the form of the function is

m-n

II (z — ar)

r=l

r=l

and, when z = <x> is a pole of order p, so that the function has m - p poles,
say cl} c.2>..., in the finite part of the plane, the form of the function is

II (Z - Or)

r=l _

m-p ~

COROLLARY IV. All the singularities of rational algebraical meromorphic
functions are accidental.

CHAPTER V.

TRANSCENDENTAL INTEGRAL FUNCTIONS.

49. WE now proceed to consider the properties of uniform functions
which have essential singularities.

The simplest instance of the occurrence of such a function has already
been referred to in § 42 ; the function has no singularity except at z = oo ,
and that value is an essential singularity solely through the failure of the
limitation to finiteness that would render the singularity accidental. The
function is then an integral function of transcendental character ; and it is
analytically represented (§ 26) by G (z) an infinite series in positive powers of
z, which converges everywhere in the finite part of the plane and acquires
an infinite value at infinity alone.

The preceding investigations shew that uniform functions, all the singu
larities of which are accidental, are rational algebraical functions — their
character being completely determined by their uniformity and the accidental
nature of their singularities, and that among such functions having the same
accidental singularities the discrimination is made, save as to a constant
factor, by means of their zeros.

Hence the zeros and the accidental singularities of a rational algebraical
function determine, save as to a constant factor, an expression of the function
which is valid for the whole plane. A question therefore arises how far
the zeros and the singularities of a transcendental function determine the
analytical expression of the function for the whole plane.

50. We shall consider first how far the discrimination of transcendental
integral functions, which have no infinite value except for z = oc , is effected
by means of their zeros*.

* The following investigations are based upon the famous memoir by Weierstrass, " Zur
Theorie der eindeutigen analytischen Functionen," published in 187G : it is included, pp. 1 — 52,
in the Abhandlungen aus der Functioiienlehre (Berlin, 1886).

In connection with the product-expression of a transcendental function, Cayley, " Memoire sur
les fonctions doublement periodiques," Liouville, t. x, (1845), pp. 385 — 420, or Collected Works,
vol. i, pp. 156 — 182, should be consulted.

50.]

CONVERGING INFINITE PRODUCTS

75

Let the zeros aly a2, a3,... be arranged in order of increasing moduli; a
finite number of terms in the series may have the same value so as to allow
for the existence of a multiple zero at any point. After the results stated
47, it will be assumed that the number of zeros is infinite ; that,

n

subject to limited repetition, they are isolated points ; and, in the present
chapter, that, as n increases indefinitely, the limit of \an\ is infinity. And it
will be assumed that at\ > 0, so that the origin is temporarily excluded from
the series of zeros.

Let z be any point in the finite part of the plane. Then only a limited
number of the zeros can lie within and on a circle centre the origin and
radius equal to \z\ ; let these be a]5 a2,..., afc_1} and let ar denote any one of
the other zeros. We proceed to form the infinite product of quantities ur,
where ur denotes

and gr is a rational integral function of z which, being subject to choice, will
be chosen so as to make the infinite product converge everywhere in the
plane. We have

00 \

w=l

a series which converges because \z < \ar\. Now let

ffr =

then

«> 1 / ^ \n
i v -1- / •* \

logi<r = - 2 -f£J ,

»j = S »4 \**rr

and therefore

Hence

•-— "

if the expression on the right-hand side be finite, that is, if the series

oo ce I / _ \ n

2 S -(-)

r=ftw=«^ \flrf

converge unconditionally. Denoting the modulus of this series by M, we
have

z
a,.

00 00 1

M < 2 2 -

r-k n=s M

SO that

sM< S 2

r=k n=s

7G

WEIERSTRASS'S CONVERGING

[50.

whence since 1 - — is the smallest of the denominators in terms of the last

«*
sum, we have

sM\l-

z

[ <

00

Z

8

1

«&

j r=k

ar

• I l

•

*-l

If, as is not infrequently the case, there be any finite integer s for which (and
therefore for all greater indices) the series

2 1

Is '

00

and therefore the series 2 \ar\-s, converges, we choose s to be that least

r=k

integer. The value of M then is finite for all finite values of z ; the series

oo co T / ~\n

2 2 - -

n

r=k

converges unconditionally and therefore
is a converging product when

Let the finite product

A-l (/ f

n |(i--

m=l l\ am

be associated as a factor with the foregoing infinite converging product. Then
the expression

oo ( f 2 \ 2

T-=I (\ ar/
is an infinite product, converging uniformly and unconditionally for all finite

00

values of z, provided the finite integer s be such as to make the series 2

converge uniformly and unconditionally.

Since the product converges uniformly and unconditionally, no product
constructed from its factors ur, say from all but one of them, can be infinite.
Now the factor

"5?i/£-Y

\ ?L\en=\n\am)

vanishes for z = am; hence f(z) vanishes for z = am. Thus the function,
evidently uniform after what has been proved, has the assigned points
Oj, a2)... and no others for its zeros.

50.]

INFINITE PRODUCT

77

Further, z = oo is an essential singularity of the function ; for it is an
essential singularity of each of the factors on account of the exponential
element in the factor.

51. But it may happen that no finite integer s can be found which will
make the series

00

r=l

converge*. We then proceed as follows.

Instead of having the same index s throughout the series, we associate
with every zero ar an integer mr chosen so as to make the series

n=l @"n \Q"n

a converging series. To obtain these integers, we take any series of decreasing
real positive quantities e, e1} e2,..., such that (i) e is less than unity and
(ii) they form an unconditionally converging series ; and we choose integers
ftir such that

These integers make the foregoing series of moduli converge. For,
neglecting the limited number of terms for which \z\^ a\, and taking e
such that

z

we have for all succeeding terms

and therefore

ar

Hence, except for the first k — 1 terms, the sum of which is finite, we have

n=k

which is finite because the series

... converges. Hence the series

n=l

s a converging series.

* For instance, there is no finite integer s that can make the infinite series
(log 2)-' + (log 3)- + (log 4)- + . . .

converge. This series is given in illustration by Hermite, Cours a la faculte des Sciences (4mc ed.
1891), p. 86.

78

WEIERSTRASS'S CONVERGING

[51.

Just as in the preceding case a special expression was formed to serve as
a typical factor in the infinite product, we now form a similar expression
for the same purpose. Evidently

1 - a; = ei<* a-*) = e
if \x\ < 1. Forming a function E (x, m) denned by the equation

m xr

S -

E (x, m)=(l-x)e r=1 r ,

we have E (x, m) =

In the preceding case it was possible to choose the integer m so that it
should be the same for all the factors of the infinite product, which was

0

ultimately proved to converge. Now, we take x = — and associate mn as
the corresponding value of m. Hence, if

/(*) =

where

< \z < |ttjfc|, we have

n=k

- s s

The infinite product represented by f(z) will converge if the double series in
the exponential be a converging series.

Denoting the double series by S, we have

\S\<* 2

2^*
2

n=kr=l

r+mn

< 2

n—k

1+TOM

1 4£

\an

on effecting the summation for r. Let A be the value of 1 —

all the remaining values of n we have

1

z !

- ; then for

— >>A,

and so

n=/fc

This series converges; hence for finite values of z\ the value of \S\ is
finite, so that S is a converging series. Hence it follows that f(z) is an

51.] INFINITE PRODUCT 79

unconditionally converging product. We now associate with f(z) as factors
the k — I functions

for i= 1, 2,..., k—1; their number being finite, their product is finite and
therefore the modified infinite product still converges. We thus have

an unconditionally converging product.

Since the product G (z) converges unconditionally, no product constructed
from its factors E, say from all but one of them, can be infinite. The factor

vanishes for the value z = an and only for this value ; hence G (z) vanishes for
z = an. It therefore appears that G(z) has the assigned points a1} a.,, a3, ...
and no others for its zeros ; and from the existence of the exponential in each
of the factors it follows that z = oo is an essential singularity of the factor and
therefore it is an essential singularity of the function.

Denoting the series in the exponential by gn (z\ so that

mn 1 / ~ \ r

*<*>-£?(£)•

71 / z \ i-. Z\

we have A — , mn = 1 e^' ;

\an / V aJ

and therefore the function obtained is

; G (z)= H \(l — — ] eg«(zl

n = l (\ Q"n,l

The series gn usually contains only a limited number of terms ; when the
number of terms increases without limit, it is only with indefinite increase
of | an | and the series is then a converging series.

It should be noted that the factors of the infinite product G (z) are the
expressions E no one of which, for the purposes of the product, is resoluble
into factors that can be distributed and recombined with similarly obtained
factors from other expressions E; there is no guarantee that the product
of the factors, if so resolved, would converge uniformly and unconditionally,
and it is to secure such convergence that the expressions E have been
constructed.

It was assumed, merely for temporary convenience, that the origin was not
a zero of the required function ; there obviously could not be a factor of
exactly the same form as the factors E if a were the origin.

80 TRANSCENDENTAL INTEGRAL FUNCTION [51.

If, however, the origin were a zero of order X, we should have merely
to associate a factor ZK with the function already constructed.

We thus obtain Weierstrass's theorem : —

It is possible to construct a transcendental integral function such that it
shall have infinity as its only essential singularity and have the origin (of
multiplicity X), a^, az, a3, ... as zeros ; and such a function is

00 ( / z\

ZK n ui — U^

n=i

where gn(z) is a rational, integral, algebraical function of z, the form of which
is dependent upon the law of succession of the zeros.

52. But, unlike uniform functions with only accidental singularities, the
function is not unique : there are an unlimited number of transcendental
integral functions with the same series of zeros and infinity as the sole essential
singularity, a theorem also due to Weierstrass.

For, if G! (z) and G (z) be two transcendental, integral functions with the
same series of zeros in the same multiplicity, and z = oo as their only essential
singularity, then

G(z}
is a function with no zeros and no infinities in the finite part of the plane.

Denoting it by £r2, then

1 ^

<72 dz

is a function which, in the finite part of the plane, has no infinities; and
therefore it can be expanded in the form

a series converging everywhere in the finite part of the plane. Choosing a
constant C0 so that 6r2 (0) = e*7", we have on integration

where g(z) = C0

and g (z) is finite everywhere in the finite part of the plane. Hence it follows
that, ifg(z) denote any integral function of z which is finite everywhere in the
finite part of the plane, and if G (z) be some transcendental integral function
with a given series of zeros and z= oo as its sole essential singularity, all
transcendental integral functions with that series of zeros and z= <x> as the
sole essential singularity are included in the form

£(*)«*».

COROLLARY I. A function which has no zeros in the finite part of the
plane, no accidental singularities and z=<x> for its sole essential singularity
is necessarily of the form

52.] AS AN INFINITE PRODUCT 81

where g (z) is an integral function of z finite everywhere in the finite part
of the plane.

COROLLARY II. Every transcendental function, which has the same zeros
in the same multiplicity as an algebraical polynomial A (z) — the number,
therefore, being necessarily finite — , ivhich has no accidental singularities and
has z = oo for its sole essential singularity, can be expressed in the form

A (z)

COROLLARY III. Every function, which has an assigned series of zeros
and an assigned series of poles and has z = oo for its sole essential singu
larity, is of the form

where the zeros of G0(z) are the assigned zeros and the zeros of Gp(z) are the
assigned poles.

For if Op (z) be any transcendental integral function, constructed as in
the proposition, which has as its zeros the poles of the required function in
the assigned multiplicity, the most general form of that function is

0p(*)e*»,

where h (z) is integral. Hence, if the most general form of function which
has those zeros for its poles be denoted by f(z), we have

f(z)Gp(z)e^

as a function with no poles, with infinity as its sole essential singularity, and
with the assigned series of zeros. But if G0 (z) be any transcendental integral
function with the assigned zeros as its zeros, the most general form of function
with those zeros is

and so f(z) Gp (z) eh ® = G0 (z) e° & ,

whence / (z) = ?$1 effW,

Lrp (z)

in which g (z) denotes g (z) — h (z).

If the number of zeros be finite, we evidently may take G0(z) as the
algebraical polynomial with those zeros as its only zeros.

If the number of poles be finite, we evidently may take Gp(z) as the
algebraical polynomial with those poles as its only zeros.

And, lastly, if a function have a finite number of zeros, a finite number
of accidental singularities and 2=00 as its sole essential singularity, it can
be expressed in the form

F.

82 PRIMARY [52.

where P and Q are rational integral polynomials. This is valid even though
the number of assigned zeros be not the same as the number of assigned
poles ; the sole effect of the inequality of these numbers is to complicate the
character of the essential singularity at infinity.

53. It follows from what has been proved that any uniform function,
having z = <x> for its sole essential singularity and any number of assigned
zeros, can be expressed as a product of expressions of the form

a

Such a quantity is called* a primary factor of the function.

It has also been proved that : —

(i) If there be no zero an, the primary factor has the form

(ii) The exponential index gn (z) may be zero for individual primary
factors, though the number of such factors must, at the utmost,
be finite f.

(iii) The factor takes the form z when the origin is a zero.
Hence we have the theorem, due to Weierstrass : —

Every uniform integral function of z can be expressed as a product of
primary factors, each of the form

(kz + I) e3W,

where g(z) is an appropriate integral function of z vanishing with z and where
k, I are constants. In particular factors, g (z) may vanish ; and either k or I,
but not both k and I, may vanish with or without a non-vanishing exponential
index g(z).

54. It thus appears that an essential distinction between transcendental
integral functions is constituted by the aggregate of their zeros : and we may
conveniently consider that all such functions are substantially the same when
they have the same zeros.

There are a few very simple sets of functions, thus discriminated by their
zeros: of each set only one member will be given, and the factor e^(z}, which
makes the variation among the members of the same set, will be neglected
for the present. Moreover, it will be assumed that the zeros are isolated
points.

I. There may be a finite number of zeros ; the simplest function is then
an algebraical polynomial.

* Weierstrass's term is Prim/unction, I.e., p. 15.

t Unless the class (§ 59) be zero, when the index is zero for all the factors.

54.] FACTORS 83

II. There may be a singly-infinite system of zeros. Various functions
will be obtained, according to the law of distribution of the zeros.

Thus let them be distributed according to a law of simple arithmetic
progression along a given line. If a be a zero, co a quantity such that co \
is the distance between two zeros and arg. co is the inclination of the line,
we have

a + mco,

for integer values of m from - oo to + oo , as the expression of the series of
the zeros. Without loss of generality we may take a at the origin — this
is merely a change of origin of coordinates — and the origin is then a
simple zero : the zeros are given by mco, for integer values of m from
— oo to + oo .

Now 2 — - = - 2 — is a diverging series ; but an integer s — the lowest

value is s = 2 — can be found for which the series S I - ] converges uni-

\mcoj
formly and unconditionally. Taking s = 2, we have

, . '-1 1 / z \n z
ffm (z) = 2 - — = — ,
»=i n vW m™
so that the primary factor of the present function is

Z \

--- )
mco/

m<a

e

and therefore, by § 52, the product

/«-,SJ(i- *-)

-oo (\ mcoj
converges uniformly and unconditionally for all finite values of z.

The term corresponding to m = 0 is to be omitted from the product ; and
it is unnecessary to assume that the numerical value of the positive infinity
for m is the same as that of the negative infinity for m. If, however, the
latter assumption be adopted, the expression can be changed into the ordinary
product-expression for a sine, by combining the primary factors due to values
of m that arc equal and opposite : in fact, then

co . TTZ
= - - sin — .

7T CO

This example is sufficient to shew the importance of the exponential term in the
primary factor. If the product be formed exactly as for an algebraical polynomial, then
the function is

z n

in the limit when both p and q are infinite. But this is known* to be

- ) - sin — .

77 0)

* Hobson's Trigonometry, § 287.

6—2

84 PRIMARY [54.

Another illustration is afforded by Gauss's II-function, which is the limit when k is
infinite of

1.2.3 ...... k

(«+!) (0+2) ...... (z+k)

This is transformed by Gauss* into the reciprocal of the expression

that is, of (1 +*) jj {(l +^) e "2l°g

the primary factors of which have the same characteristic form as in the preceding
investigation, though not the same literal form.

It is chiefly for convenience that the index of the exponential part of the primary

t-l 1/2 \n

factor is taken, in § 50, in the form 2 - ( — ) . With equal effectiveness it may be

n=l % \~^T /
»-l 1

taken in the form 2 - br nzn. provided the series
'

r=k «=i n
converge uniformly and unconditionally.

Ex. 1. Prove that each of the products

form=+l, ±3, +5, ...... to infinity, and

the term for n = Q being excluded from the latter product, converges uniformly and uncon
ditionally and that each of them is equal to cos z. (Hermite and Weyr.)

Ex. 2. Prove that, if the zeros of a transcendental integral function be given by the
series

0) +&>, ±4w, +9cB, ...... to infinity,

the simplest of the set of functions thereby determined can be expressed in the form

( fz\*\ , (. fz\*\
sm X?r I - }- sin -UTT - ) }- .
I W ) ( W J

Ex. 3. Construct the set of transcendental integral functions which have in common
the scries of zeros determined by the law m2a>l + 2m<a2 + a>3 for all integral values of m
between - oo and + oo ; and express the simplest of the set in terms of circular functions, j

55. The law of distribution of the zeros, next in importance and sub
stantially next in point of simplicity, is that in which the zeros form a doubly-
infinite double arithmetic progression, the points being the oo 2 intersections
of one infinite system of equidistant parallel straight lines with another
infinite system of equidistant parallel straight lines.

The origin may, without loss of generality, be taken as one of the zeros.
If a) be the coordinate of the nearest zero along the line of one system
passing through the origin, and &>' be the coordinate of the nearest zero along

* Ges. Wcrke, t. Hi, p. 145; the example is quoted in this connection by Weierstrass, I.e., !
p. 15.

55.] FACTIOUS 85

the line of the other system passing through the origin, then the complete
series of zeros is given by

fl = mw + mm,

for all integral values of m and all integral values of ni between — <x> and
+ oo . The system of points may be regarded as doubly -periodic, having &>
arid &>' for periods.

It must be assumed that the two systems of lines intersect. Other
wise, w and to' would have the same argument and their ratio would be a real
quantity, say a ; and then

ft

— = m + m a.

CO

Whether a be commensurable or incommensurable, the number of pairs
of integers, for which m + in' a. is zero or may be made less than any small
quantity 8, is infinite ; and in either case we should have the origin a zero
for each such pair, that is, altogether the origin would be a zero of infinite
multiplicity. This property of a function is to be considered as excluded,
for it would make the origin an essential singularity instead of, as required,
an ordinary point of the transcendental integral function. Hence the ratio of
the quantities w and w' is not real.

56. For the construction of the primary factor, it is necessary to render
the series

converging, by appropriate choice of integers sm>m. It is found to be
possible to choose an integer s to be the same for every term of the series,
corresponding to the simpler case of the general investigation, given in § 50.
As a matter of fact, the series

diverges for s = I (we have not made any assumption that the positive and
the negative infinities for m are numerically equal, nor similarly as to m') ;
the series converges for s = 2, but its value depends upon the relative values
of the infinities for m and m'; and s = 3 is the lowest integral value for which,
as for all greater values, the series converges uniformly and unconditionally.

There are various ways of proving the uniform and unconditional conver
gence of the series 2ft~M when /* > 2 : the following proof is based upon a
general method due to Eisenstein*.

»I=QO n=oo

First, the series S 2 (m2 + n*)~* converges uniformly and uricondi-

m= — «> n= -oo

tionally, if /j,> 1. Let the series be arranged in partial series : for this purpose,

Crelle, t. xxxv, (1847), p. 161 ; a geometrical exposition is given by Halphen, Traite des
fonctions elliptiques, t. i, pp. 358 — 362.

86 WEIERSTRASS'S FUNCTION AS [56.

we choose integers k and I, and include in each such partial series all
the terms which satisfy the inequalities

m ^ 2*+1,

so that the number of values of m is 2* and the number of values of n is 2*.
Then, if k + I = %K, we have

so that each term in the partial series ^ ^- . The number of terms in the

^" J*

partial series is 2fc . 2*, that is, 22K : so that the sum of the terms in the
partial series is

Take the upper limit of k and I to be p, ultimately to be made infinite.
Then the sum of all the partial series is

which, when p = oo , is a finite quantity if p > 1.
Next, let (a = a. + /3i, «' = 7 + Si, so that

ft = mw + nay' = ma + ny + i (m{3 + n8) ;
hence, if 6 =• ma. + nj, (j> = m(3 + n$,

we have | ft 2 = fr + </>2.

Now take integers r and s such that

r<0<r + \, s<(jxs + ~L.

The number of terms ft satisfying these conditions is definitely finite and is
independent of m and n. For since

m(«S —

n a -

and a8 — (3y does not vanish because o>'/a> is not purely real, the number of
values of in is the integral part of

(r + 1)8 — sy

a.8 — fiy
less the integral part of

r8 — (s + 1 ) 7
a.8 — fly

that is, it is the integral part of (7 + 8)/(«8 — #7). Similarly, the number of
values of n is the integral part of (a + /3)/(aS - j3j). Let the product of the

56.] A DOUBLY-INFINITE PRODUCT 87

last two integers be q ; then the number of terms fl satisfying the in
equalities is q.

Then 22 1 ft \~* = 22 (&> + p)~*

< q 22 (r2 + s2)-'*,
which, by the preceding result, is finite when yu,> 1. Hence

22 (mco + m'(»)'}~-»-

converges uniformly and unconditionally when //, > 1 ; and therefore the least
value of s, an integer for which

22 (mco + m'co')~s
converges uniformly and unconditionally, is 3.

The series 22(?tto) + m'<»')~2 has a finite sum, the value of which depends* upon
the infinite limits for the summation with regard to m and m'. This dependence is
inconvenient and it is therefore excluded in view of our present purpose.

Ex. Prove in the same manner that the series

the multiple summation extending over all integers mlt m2, ...... , mn between — oo and

+ oo , converges uniformly and unconditionally if 2/j.>n. (Eiseustein.)

57. Returning now to the construction of the transcendental integral
function the zeros of which are the various points H, we use the preceding
result in connection with § 50 to form the general primary factor. Since
s = 3, we have

s-l

and therefore the primary factor is

Moreover, the origin is a simple zero. Hence, denoting the required function
by a (z), we have

00 °°

<r(z) = zU H

— 00 -00

as a transcendental integral function which, since the product converges uni
formly and unconditionally for all finite values of z, exists and has a finite
value everywhere in the finite part of the plane; the quantity O denotes
mco + mV, and the double product is taken for all values of m and of m
between — oo and + oo , simultaneous zero values alone being excluded.

This function will be called Weierstrass's o-function ; it is of importance
in the theory of doubly-periodic functions which will be discussed in Chapter
XL

* See a paper by the author, Quart. Journ. of Math., vol. xxi, (1886), pp. 261—280.

88 PRIMARY FACTORS [57.

Ex. If the doubly-infinite series of zeros be the points given by

Q = m2^ + 2wm&>2 + «2o>3,

wi> W2) W3 being such complex constants that i2 does not vanish for real values of m and n,
then the series

2 2 Q-*

converges for s = 2. The primary factor is thus

and the simplest transcendental integral function having the assigned zeros is

The actual points that are the zeros are the intersections of two infinite systems of
parabolas.

58. One more result — of a negative character — will be adduced in this
connection. We have dealt with the case in which the system of zeros is a
singly-infinite arithmetical progression of points along one straight line and
with the case in which the system of zeros is a doubly-infinite arithmetical
progression of points along two different straight lines : it is easy to see that
a uniform transcendental integral function cannot exist with a triply -infinite
arithmetical progression of points for zeros.

A triply-infinite arithmetical progression of points would be represented
by all the possible values of

for all possible integer values for p1} p.,, p3 between — oo and + oc , where no
two of the arguments of the complex constants flj, H2, O3 are equal. Let

tlr = o)r + i(or', (r = 1, 2, 3) ;

then, as will be proved (§ 107) in connection with a later proposition, it is
possible* — and possible in an unlimited number of ways — to determine
integers plt p-2,ps so that, save as to infinitesimal quantities,

Pi _ _ £2 ___ PS

all the denominators in which equations differ from zero on account of the
fact that no two arguments of the three quantities fl1} H2, Ha are equal. For
each such set of determined integers we have

&.Qi+p&+p»to»

zero or infinitesimal, so that the origin is a zero of unlimited multiplicity or,
in other words, there is a space at the origin containing an unlimited number
of zeros. In either case the origin is an essential singularity, contrary to

* Jacobi, Oes. Werke, t. ii, p. 27.

58.] CLASS OF A FUNCTION 89

the hypothesis that the only essential singularity is for z — oo ; and hence a
uniform transcendental function cannot exist having a triply-infinite arith
metical succession of zeros.

59. In effecting the formation of a transcendental integral function by
means of its primary factors, it was seen that the expression of the primary
factor depends upon the values of the integers which make

a converging series. Moreover, the primary factors are not unique in form,
because any finite number of terms of the proper form can be added to the
exponential index in

and such terms will only the more effectively secure the convergence of the
infinite product. But there is a lower limit to the removal of terms with the
highest exponents from the index of the exponential ; for there are, in general,
minimum values for the integers m1} m»,..., below which these integers can
not be reduced, if the convergence of the product is to be secured.

The simplest case, in which the exponential must be retained in the
primary factor in order to secure the convergence of the infinite product, is
that discussed in § 50, viz., when the integers ml, w2)... are equal to one
another. Let m denote this common value for a given function, and let
m be the least integer effective for the purpose : the function is then said*
to be of class m, and the condition that it should be of class m is that the
integer m be the least integer to make the series

converge uniformly and unconditionally, the constants a being the zeros of
the function.

Thus algebraical polynomials are of class 0 ; the circular functions sin z
and cos z are of class 1 ; Wcierstrass's o--function, and the Jacobian elliptic
function sn z are of class 2, and so on : but in .no one of these classes do the
functions mentioned constitute the whole of the functions of that class.

60. One or two of the simpler properties of an aggregate of transcen
dental integral functions of the same class can easily be obtained.

Let a function f(z), of class n, have a zero of order r at the origin and

* The French word is genre ; the Italian is genere. Laguerre (see references on p. 92) appears
to have been the first to discuss the class of transcendental integral functions.

90

CLASS-PROPERTIES OF

[60.

have «!, a2)... for its other zeros, arranged in order of increasing moduli.
Then, by § 50, the function /O) can be expressed in the form

(*)='

M 1 / £\8

where </; (V) denotes the series 2 -f— 1 and G(z) must be properly deter
mined to secure the equality.

Now the series

is one which converges uniformly for all values of z that do not coincide with
one of the points a, that is, with one of the zeros of the original function.
For the sum of the series of the moduli of its terms is

1

Let d be the least of the quantities

1

, necessarily non-evanescent be

cause z does not coincide with any of the points a ; then the sum of the series

IS 1

which is a converging series since the function is of class n. Hence the
series of moduli converges and therefore the original series converges ; let it
be denoted by S (z), so that

1

=2

We have

Each step of this process is reversible in all cases in which the original pro-

f (z\
duct converges; if, therefore, it can be shewn of a function f(z) that -rr4

takes this form, the function is thereby proved to be of class n.
If there be no zero at the origin, the term - is absent.

CO.] TRANSCENDENTAL INTEGRAL JUNCTIONS 91

If the exponential factor G(z) be a constant so that G' (z) is zero, the
function /(.z) is said to be a simple function of class n.

61. There are one or two criteria to determine the class of a function :
the simplest of them is contained in the following proposition, due to
Laguerre*.

If, as z tends to the value <x> , a very great value of z can be found for

f'(z\
which the limit of z~n --jr\ , where f (z) is a transcendental, integral function,

J\z)
tends uniformly to the value zero, then f (z} is of class n.

Take a circle centre the origin and radius R, equal to this value of \z\\
then, by § 24, II., the integral

f'(t) dt

JL/lo!

SvtJ *»/(*)

taken round the circle, is zero when R becomes indefinitely great. But the
value of the integral is, by the Corollary in § 20,

' (t) 6A

+

!_ f<*> J./'_(0 Jfc_ _L y (
27ri J V- f(t) t-z 2-n-i <=1 J

tn f(fi t-Z 2-7TI J tn f(t) t-Z 2lri i=i J tn f(t} t-z'

taken round small circles enclosing the origin, the point z, and the points
a,i, which are the infinities of the subject of integration; the origin being
supposed a zero of /(t) of multiplicity r.

1 f» !/'(*) dt ._!/'(*)

JMOW

tnf(t}t-Z Znf(2}'
dt I I

»/ \^ /•

Shr»,

1 fWlf(t
iriJ «"/(0

L f<0> 1£(Q _^_ <^>(^) r
SwtJ tnf(t)t-z zn zn+*'

where ^> (^) denotes the integral, algebraical, polynomial

V " f +0 j~ i -f ~ if +•••'

when t is made zero. Hence

and therefore

which, by § GO, shews that/(V) is of class n.

* Comptcs Rendus, t. xciv, (1882), p. G36.

92 CLASS-PROPERTIES OF [61.

COROLLARY. The product of any finite number of functions of the same
class n is a function of class not higher than n ; and the class of the product
of any finite number of functions of different classes is not greater than the
highest class of the component functions.

The following are the chief references to memoirs discussing the class of functions :

Laguerrc, Comptes Rendus, t. xciv, (1882), pp. 160-163, pp. 635—638, ib. t. xcv, (1882),
pp. 828—831, ib. t. xcviii, (1884), pp. 79—81 ;

Poincare, Bull, des Sciences Math., t. xi, (1883), pp. 136—144 ;

Cesaro, Comptes Rendm, t. xcix, (1884), pp. 26—27, followed (p. 27) by a note by
Hermite; Giornale di Battaglini, t. xxii, (1884), pp. 191 — 200;

Vivanti, Giornale di Battaglini, t. xxii, (1884), pp. 243—261, pp. 378—380, ib. t. xxiii,
(1885), pp. 96—122, ib. t. xxvi, (1888), pp. 303—314 ;

Hermite, Cours d la faculte' des Sciences (4me ed., 1891), pp. 91 — 93.

Ex. 1. The function

2
1=1

where the quantities c are constants, n is a finite integer, and the functions J\ (z) are
algebraical polynomials, is of class unity.

Ex. 2. If a simple function be of class %, its derivative is also of class n.

Ex. 3. Discuss the conditions under which the sum of two functions, each of class n,
is also of class n.

Ex. 4. Examine the following test for the class of a function, due to Poincare.

Let a be any number, no matter how small provided its argument be such that eaz
vanishes when z tends towards infinity. Then / (z) is of class n, if the limit of

vanish with indefinite increase of z.

A possible value of a is 2 ciai~n~1, where C; is a constant of modulus unity.

Ex. 5. Verify the following test for the class of a function, due to de Sparre*.

Let X be any positive non-infinitesimal quantity ; then the function / (z) is of class n,
if the limit, for m = oo , of

\amn~l{\am + i\-\am\}
be not less than X. Thus sin z is of class unity.

Ex. 6. Let the roots of 0n + 1 = l be 1, a, a2, ...... , an; and let f (s) be a function

of class n. Then forming the product

n/(a«4

we evidently have an integral function of zn + 1; let it be denoted by F(zn + 1). The roots of
* Comptes Rendus, t. cii, (1886), p. 741.

61.] TRANSCENDENTAL INTEGRAL FUNCTIONS 93

F(zn+l) = Q are a^'for i=l, 2, and s = 0, 1, , n\ and therefore, replacing zn + 1 by z,

the roots ofF(z) = 0 are a?*1 for i=l, 2, .......

Since/ (z) is of class n, the series

converges uniformly and unconditionally. This series is the sum of the first powers of the
reciprocals of the roots of F(z}~ 0; hence, according to the definition (p. 89), F(z) is of
class zero.

It therefore follows that from, a function of any class a function of class zero with a
modified variable can be deduced. Conversely, by appropriately modifying the variable of
a given function of class zero, it is possible to deduce functions of any required class.

Ex. 7. If all the zeros of the function

=1 r anr
\
be real, then all the zeros of its derivative are also real. (Witting.)

00 I / ~ \

U\(l--)e'
«=* ^\ «W

CHAPTER VI.

FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES.

62. SOME indications regarding the character of a function at an
essential singularity have already been given. Thus, though the function

is regular in the vicinity of such a point a, it may, like sn - at the origin,

%

have a zero of unlimited multiplicity or an infinity of unlimited multiplicity
at the point ; and in either case the point is such that there is no factor of
the form (z — a)x which can be associated with the function so as to make the
point an ordinary point for the modified function. Moreover, even when
the path of approach to the essential singularity is specified, the value
acquired is not definite : thus, as z approaches the origin along the axis of x,

so that its value may be taken to be 1 -f- (4>mK + x), the value of sn - is not

z

definite in the limit when m is made infinite. One characteristic of the
point is the indefiniteness of value of the function there, though in the
vicinity the function is uniform.

A brief statement and a proof of this characteristic were given in § 33 ;
the theorem there proved — that a uniform analytical function can assume
any value at an essential singularity — may also be proved as follows. The
essential singularity will be taken at infinity — a supposition that will be
found not to detract from generality.

Let f(z) be a function having any number of zeros and any number
of accidental singularities and £ = oo for its sole essential singularity ; then
it can be expressed in the form

/w-88*"'

where G1 (z) is algebraical or transcendental according as the number of zeros
is finite or infinite and G2(z) is algebraical or transcendental according as
the number of accidental singularities is finite or infinite.

If Cr2 (z) be transcendental, we can omit the generalising factor e°(z).
Then f(z) has an infinite number of accidental singularities ; each of them
in the finite part of the plane is of only finite multiplicity and therefore some
of them must be at infinity. At each such point, the function G2 (z) vanishes
and Ol (z) does not vanish ; and so f(z) has infinite values for z = oo .

62.] VALUE AT AN ESSENTIAL SINGULARITY 95

If Gz (2) be algebraical and Gl (z) be also algebraical, then the factor ea(z)
may not be omitted, for its omission would make f(z) an algebraical function.
Now z = oo is either an ordinary point or an accidental singularity of

ft <*)/<?.<*);

hence as g (z} is integral there are infinite values of z which make

infinite.

If G.>.(z) be algebraical and G^ (z) be transcendental, the factor eg(z) maybe
omitted. Let al5 a2,..., an be the roots of G2(z): then taking

f(z)= ^-

we have Ar=

a non-vanishing constant ; and so

where Gn (z) is a transcendental integral function. When 2 = oo , the value
of G3(z)/G.,(z) is zero, but Gn(z) is infinite ; hence f(z) has infinite values for

Z= 00 .

Similarly it may be shewn, as follows, that/(z) has zero values for 0 = oo .

In the first of the preceding cases, if Gl (z) be transcendental, so that f (z)
has an infinite number of zeros, then some of them must be at an infinite
distance; f(z) has a zero value for each such point. And if GI(Z) be
algebraical, then there are infinite values of z which, not being zeros of
G2(z), make f(z) vanish.

In the second case, when z is made infinite with such an argument as to
make the highest term in g(z) a real negative quantity, then f(z) vanishes
for that infinite value of z.

In the third case,/(V) vanishes for a zero of G1(z) that is at infinity.

Hence the value of f (z) for z= oo is not definite. If, moreover, there
be any value neither zero nor infinity, say G, which f(z) cannot acquire
for z = oo , then

/(*)-C

is a function which cannot be zero at infinity and therefore all its zeros are
in the finite part of the plane : no one of them is an essential singularity, for
f(z) has only a single value at any point in the finite part of the plane; hence
they are finite in number and are isolated points. Let H1 (z) be the alge
braical polynomial having them for its zeros. The accidental singularities
of f(z} — C are the accidental singularities of f(z) ; hence

96 FORM OF A FUNCTION NEAR [62.

where, if G2(z) be algebraical, the exponential h(z) must occur, since f(z),
and therefore f(z) — C, is transcendental. The function

-| sy / \

TJ1 ( ~\ — _ 2 \ / 0—h (z)

•* \*/ f t ~\ n TT f n\

J(z)-L H1 (z)

evidently has z= oo for an essential singularity, so that, by the second or
the third case above, it certainly has an infinite value for z = co , that is,
f(z) certainly acquires the value G for z= GO .

Hence the function can acquire any value at an essential singularity.

63. We now proceed to obtain the character of the expression of a
function at a point z which, lying in the region of continuity, is in the
vicinity of an essential singularity b in the finite part of the plane.

With b as centre describe two circles, so that their circumferences and
the whole area between them lie entirely within the region of continuity.
The radius of the inner circle is to be as small as possible consistent with
this condition; and therefore, as it will be assumed that b is the only
singularity in its own immediate vicinity, this radius may be made very
small.

The ordinary point z of the function may be taken as lying within the
circular ring-formed part of the region of continuity. At all such points in
this band, the function is holomorphic ; and therefore, by Laurent's Theorem
(§ 28), it can be expanded in a converging series of positive and negative
integral powers of z — b in the form

+ V-L(Z — 6)"1 + v2 (z — 6)~2 + . . .,
where the coefficients un are determined by the equation

un =

the integrals being taken positively round the outer circle, and the coefficients
vn are determined by the equation

the integrals being taken positively round the inner circle.

The series of positive powers converges everywhere within the outer circle
of centre b, and so (§ 26) it may be denoted by P (z - b) ; and the function P
may be either algebraical or transcendental.

The series of negative powers converges everywhere without the inner
circle of centre b ; and, since 6 is not an accidental but an essential singularity
of the function, the series of negative powers contains an infinite number of

63.] AN ESSENTIAL SINGULARITY 97

terms. It may be denoted by G I -- rh a series converging for all points

\z — o/

in the plane except z = b and vanishing when z — b = co.

Thus

is the analytical representation of the function in the vicinity of its essential
singularity b ; the function G is transcendental and converges everywhere in
tlie plane except at z =• b, and the function P, if transcendental, converges
uniformly and unconditionally for sufficiently small values of | z — b \ .

Had the singularity at b been accidental, the function G would have been
algebraical.

COROLLARY I. If the function have any essential singularity other than
b, it is an essential singularity of P (z — b) continued outside the outer circle ;

but it is not an essential singularity of G ( -- j] , for the latter function

\z — ol

converges everywhere in the plane outside the inner circle.

COROLLARY II. Suppose the function has no singularity in the plane
except at the point b ; then the outer circle can have its radius made infinite.
In that case, all positive powers except the constant term w0 disappear:
and even this term survives only in case the function have a finite value at
infinity. The expression for the function is

and the transcendental series converges everywhere outside the infinitesimal
circle round b, that is, everywhere in the plane except at the point b. Hence
the function can be represented by

This special result is deduced by Weierstrass from the earlier investiga
tions*, as follows. If f(z) be such a function with an essential singularity at
b, and if we change the independent variable by the relation

Z/==^b'

thcn/(V) changes into a function of z', the only essential singularity of which
is at / = GO . It has no other singularity in the plane ; and the form of the
function is therefore G (z'), that is, a function having an essential singularity
at b but no other singularity in the plane is

* Weierstrass (I.e.), p. 27.
F.

98 FORM OF A FUNCTION NEAR [63.

COROLLARY III. The most general expression of a function having its
sole essential singularity at b a point in the finite part of the plane and any
number of accidental singularities is

G,

where the zeros of the function are the zeros of GI, the accidental singularities
of the function are the zeros of G2, and the function g in the exponential is a
function which is finite everyiuhere except at b.

This can be derived in the same way as before ; or it can be deduced
from the corresponding theorem relating to transcendental integral functions,
as above. It would be necessary to construct an integral function G2(z')
having as its zeros

and then to replace z by - — j ; and G., is algebraical or transcendental,

Z 0

according as the number of zeros is finite or infinite.
Similarly we obtain the following result :

COROLLARY IV. A uniform function of z, which has its sole essential
singularity at b a point in the finite part of the plane and no accidental
singularities, can be represented in the form of an infinite product of primary
factors of the form

\z — b

which converges uniformly and unconditionally everywhere in the plane except
at z = b.

The function g ( =] is an integral function of T vanishing when

J \z-bj z-b

r vanishes; and k and I are constants. In particular factors, q( T)

z - b ^ \z - b)

may vanish ; and either k or I (but not both k and I) may vanish with or

without a vanishing exponent q { T ) .

J \z-bj

If tt{ be any zero, the corresponding primary factor may evidently be
expressed in the form

(z —

,z —

Similarly, for a uniform function of z with its sole essential singularity at b and
any number of accidental singularities, the product-form is at once derivable

63.] AN ESSENTIAL SINGULARITY 99

by applying the result of the present Corollary to the result given in
Corollary III.

These results, combined with the results of Chapter V., give the complete
general theory of uniform functions with only one essential singularity.

64. We now proceed to the consideration of functions, which have a
limited number of assigned essential singularities.

The theorem of § 63 gives an expression for the function at any point in
the band between the two circles there drawn.

Let c be such a point, which is thus an ordinary point for the function ;
then in the domain of c, the function is expansible in a form Pl (z — c).
This domain may extend to an essential singularity b, or it may be limited
by a pole d which is nearer to c than b is, or it may be limited by an
essential singularity / which is nearer to c than b is. In the first case, we
form a continuation of the function in a direction away from b; in the
second case, we continue the function by associating with the function
a factor (z — d)n which takes account of the accidental singularity ; in
the third case, we form a continuation of the function towards f. Taking
the continuations for successive domains of points in the vicinity of/, we can
obtain the value of the function for points on two circles that have / for
their common centre. Using these values, as in § 63, to obtain coefficients,
we ultimately construct a series of positive and negative powers converging
except at / for the vicinity of/ Different expressions in different parts
of the plane will thus be obtained, each being valid only in a particular
portion: the aggregate of all of them is the analytical expression of the
function for the whole of the region of the plane where the function exists.

We thus have one mode of representation of the function ; its chief
advantage is that it indicates the form in the vicinity of any point, though it
gives no suggestion of the possible modification of character elsewhere. This
deficiency renders the representation insufficiently precise and complete ; and
it is therefore necessary to have another mode of representation.

65. Suppose that the function has n essential singularities a,!, a*,..., an
and that it has no other singularity. Let a circle, or any simple closed
curve, be drawn enclosing them all, every point of the boundary as well
as the included area (with the exception of the n singularities) lying in
the region of continuity of the function.

Let z be any ordinary point in the interior of the circle or curve ; and
consider the integral , f/+\

I '*=-***>

taken round the curve. If we surround z and each of the n singularities by
small circles with the respective points for centres, then the integral round

7—2

100 FUNCTIONS WITH A LIMITED NUMBER [65.

the outer curve is equal to the sum of the values of the integral taken round
the n + l circles. Thus

and therefore

The left-hand side of the equation isf(z).
Evaluating the integrals, we have

where Gr is, as before, a transcendental function of - - — vanishing when

1

is zero.

z — ar

Now, of these functions, Gr{- -] converges everywhere in the plane

\& \jbfj

except at ar : and therefore, as n is finite,

r=i \z - a
is a function which converges everywhere in the plane except at the n points

Clj , . . . , an .

Because z = oc is not an essential singularity of f(z), the radius of the

circle in the integral =—. . ! /-- dt may be indefinitely increased. The value

ZTTI J s t — z

of f(t) tends, with unlimited increase of t, to some determinate value G which
is not infinite ; hence, as in § 24, II., Corollary, the value of the integral is
C. We therefore have the result that/0) can be expressed in the form

\z-a,
or, absorbing the constant C into the functions G and replacing the limitation,

that the function Gr(— — } shall vanish for — = 0, by the limitation

\z — arj z — ar

that, for the same value =0, it shall be finite, we have the theorem*:—

z — ar

If a given function f(z) have n singularities a^,..., an, all of which are in
the finite part of the plane and are essential singularities, it can be expressed
in the form

2G f-M,

r=i r \z - aj '

* The method of proof, by an integration, is used for brevity : the theorem can be established
by purely algebraical reasoning.

65.] OF ESSENTIAL SINGULARITIES 101

where Gr is a transcendental function converging everywhere in the plane

except at ar and having a determinate finite value gr for - — = 0, such

z — cir

n

that 2 gr is the finite value of the given function at infinity.
r=l

COROLLARY. If the given function have a singularity at oo , and n singu
larities in the finite part of the plane, then the function can be expressed in
the form

w / 1 \
G(z) + SG,(— L-J,

r=i \z-arr

where Gr is a transcendental or an algebraic polynomial function, according
as ar is an essential or an accidental singularity : and so also for G (z), accord
ing to the character of the singularity at infinity.

66. Any uniform function, which has an essential singularity at z = a,
can (§ 63) be expressed in the form

for points z in the vicinity of a. Suppose that, for points in this vicinity,
the function f(z) has no zero ; that it has no accidental singularity ; and
therefore, among such points z, the function

1 df(z)
/(*) dz

has no pole, and therefore no singularity except that at a which is essential.
Hence it can be expanded in the form

G(^+P(z-a\

z-a

where G converges everywhere in the plane except at a, and vanishes for
= 0. Let

z — a dz

, / 1 \

where 0^ I ^— ^ I converges everywhere in the plane except at a, and vanishes

for — — = o.

z — a

Then c, evidently not an infinite quantity, is an integer. To prove this,
describe a small circle of radius p round a : then taking z-a = pe91 so that

— = idd, we have
z — a

l M(*\

dz = P (z — a) dz

102 FUNCTIONS WITH A LIMITED NUMBER [66.

and therefore

Now JP(z — a)dz is a uniform function : and so is f(z). But a change
of 6 into 6 + 2-7T does not alter z or any of the functions : thus

actotr — 1 •

~~ *• i

and therefore c is an integer.

67, If the function /(z) have essential singularities alt..., an and no
others, then' it can be expressed in the form

n /I

C+ $9J-±

r=i \z-ar

If there be no zeros for this function f(z) anywhere (except of course such
as may enter through the indeterminateness at the essential singularities),
then

/(*) dz

has n essential singularities a1}..., an and no other singularities of any kind.
Hence it can be expressed in the form

n / 1 \
C+ 2 Gr(- -),
r=i \z-a,rl

where the function Gr vanishes with . Let

z — ar

cr d

I T~~

\js — a,./ z — ar dz { r\z — ar

where Gr I ) is a function of the same kind as Gr ( ) .

\z — ar/ \z — arj

Then all the coefficients cr, evidently not infinite quantities, are integers.
For, let a small circle of radius p be drawn round ar : then, if z — ar = peei, we
have

crdz

z — ar

= cri6,

and — ^ — = dPs (z - ar).

z — as

We proceed as before : the expression for the function in the former
case is changed so that now the sum 2Pg(0— ar) for 5 = !,..., i — 1,
r + 1,..., n is a uniform function; there is no other change. In exactly the
same way as before, we shew that every one of the coefficients cr is an
integer.

Hence it appears that if a given function f(z) have, in the finite part of

67.] OF ESSENTIAL SINGULARITIES 103

the plane, n essential singularities al,..., an and no other singularities and if
it have no zeros anywhere in the plane, then

f(z) dz

where all the coefficients c* are integers, and the functions G converge every
where in the plane except at the essential singularities and Gi vanishes for

-J-- 0.

Now, since f(z) has no singularity at oo , we have for very large values of z

and /'W = _>_

Z*

and therefore, for very large values of z,

_

f(z) dz u0 z2 z3

Thus there is no constant term in =7-^ ^r-^ , and there is no term in -. But

/(*) dz z

the above expression for it gives G as the constant term, which must therefore

vanish ; and it gives 2c; as the coefficient of - , for -7- •< (r< ( - — H will begin

z dz [ \z — ftj/ J

with — at least ; thus ^a must therefore also vanish.

Z"

Hence for a function f (z) which has no singularity at z= oo and no
zeros anywhere in the plane and of which the only singularities are the n
essential singularities at a1} a2,..., an, we have

/ (z) dz i=i z - Oi i=i dz ( \z- a
where the coefficients a are integers subject to the condition

n

2 ct = 0.

i=l

If an= oo , so that 2= GO is an essential singularity in addition to a2, a2,...,
an_j, there is a term 6r (z) instead of Gn( — - ] ; there is no term, that corre-

\Z — C^n/
/^

spends to - — , but there may be a constant G. Writing

—

z —

with the condition that G (z) vanishes when z — 0, we then have

-
= __ g ^

i=iz-at dz( v /J ».

104 PRODUCT-EXPRESSION OF [67.

where the coefficients d are integers, but are no longer subject to the
condition that their sum vanishes.

Let R* (z} denote the function

the product extending over the factors associated with the essential sin
gularities of f(z) that lie in the finite part of the plane; thus R*(z) is a
rational algebraical rneromorphic function. Since

1 dR*(z) = 2 d
R* (z) dz ~ i=\z — a,i'
we have

1 df(z) _ 1 dR*(z) =$ d_(-Q ( *
f(z) dz R* (z) dz i=\dz\ l\z — a^

where Gn ( — - ) is to be replaced by G (z) if an = <x> , that is, if z — oo be an
\z — anj

essential singularity off(z). Hence, except as to an undetermined constant
factor, we have

t=i

which is therefore an analytical representation of a function with n essential
singularities, no accidental singularities, and no zeros: and the rational alge
braical function R* (z) becomes zero or oo only at the singularities off(z).

If z = oo be not an essential singularity, then R* (z) for z = oo is equal to

M

unity because 2 Cf = 0.
1=1

COROLLARY. It is easy to see, from § 43, that, if the point a; be only an
accidental singularity, then a is a negative integer and wj I — - ) is zero : so

\Z — Oii/

that the polar property at c^ is determined by the occurrence of a factor
(z — a{)Ci solely in the denominator of the rational meromorphic function R* (z).

And, in general, each of the integral coefficients a is determined from the
expansion of the function f'(z) +f(z) in the vicinity of the singularity
with which it is associated.

68. Another form of expression for the function can be obtained from
the preceding; and it is valid even when the function has zeros not
absorbed into the essential singularities f.

Consider a function with one essential singularity, and let a be the
point ; and suppose that, within a finite circle of centre a (or within a finite
simple curve which encloses a), there are m simple zeros a, /3,..., X of the

+ See Guichard, TMorie des points singuliers essentiels, (These, Gauthier-Villars, Paris, 1883),
especially the first part.

68.] A FUNCTION 105

function f(z) — m being assumed to be finite, and it being also assumed that
there are no accidental singularities within the circle. Then, if

/(*) = (* - «) (z - /3). . .(* -\)F (z\

the function F (z) has a for an essential singularity and has no zeros within
the circle. Hence, for points z within the circle,

where (?, ( ----- ) converges everywhere in the plane and vanishes with - — ,

\z — a] z a

and P(z — a) is an integral function converging uniformly and unconditionally
within the circle ; moreover, c is an integer. Thus

F(z) = A(z- a)" eGl
Let (*-a)(*-/3)...(*-X) = (*-a

_ ( y _ r, \m

a)

then f(z) = (* - dTffl -- F(z}

\z u/

= A(z- ar+°gi (~}eG> ^ e ^~a] "z .
\z — ft/

Now of this product- expression for/(V) it should be noted: —

(i) That m + c is an integer, finite because m and c are finite :

<?,—-
(ii) The function e ' ^z~a' can be expressed in the form of a series con

verging uniformly and unconditionally everywhere, except at z = a, and

proceeding in powers of - — in the form

z a

....

z — a (z — af

It has no zero within the circle considered, for F (z) has no zero. Also gl(- - 1

\z a/

algebraical function of — ' — , beginning with unity and containing only

is an

z — a

a finite number of terms : hence, multiplying the two series together, we have

as the product a series proceeding in powers of - in the form

£ ~" a

— a

which converges uniformly and unconditionally everywhere outside any small
circle round a, that is, everywhere except at a. Let this series be denoted by

106 PRODUCT-EXPRESSION OF [68.

H I ]; it has an essential singularity at a and its only zeros are the

\z-aj

points a, (3,..., X, for the series multiplied by gl (— -) has

\z — ft/

no zeros :

(iii) The function fP (z — a) dz is a series of positive powers of z — a,
converging uniformly in the vicinity of a; and therefore Q^(z-d)dz can ke
expanded in a series of positive integral powers of z — a which converges
in the vicinity of a. Let it be denoted by Q (z — a) which, since it is a
factor of F (z), has no zeros within the circle.

Hence we have

/(*) = A (z - aYQ (z - a) H
where p, is an integer ; H ( — - J is a series that converges everywhere except

at a, is equal to unity when - vanishes, and has as its zeros the (finite)

z — a

number of zeros assigned to f(z) within a finite circle of centre a ; and
Q (z — a) is a series of positive powers of z — a beginning with unity which
converges — but has no zero — within the circle.

The foregoing function f(z) is supposed to have no essential singularity
except at ft. If, however, a given function have singularities at points
other than a, then the circle would be taken of radius less than the distance
of a from the nearest essential singularity.

Introducing a new function f{ (z} defined by the equation

the value of /[ (z) is Q (z — a) within the circle, but it is not determined by
the foregoing analysis for points without the circle. Moreover, as (z — a)*

and also Hi— ] are finite everywhere except possibly at a, it follows

that essential singularities of f(z) other than a must be essential singu
larities of fj (z). Also since /i (z) is Q(z — a) in the immediate vicinity of a,
this point is not an essential singularity of /i (z).

Thus /i (z) is a function of the same kind as f(z) ; it has all the essential
singularities of f(z) except ft, but it has fewer zeros, on account of the m

zeros of f(z) possessed by H ( - — ] . The foregoing expression for f(z) is

\Z — ft/

the one referred to at the beginning of the section.

If we choose to absorb into /x (z} the factors e \z~a' and e?P(z~° dz,
which occur in

(z - $*• ffl f Jil ^ (T- a)

\2 — ft/

68.] A FUNCTION 107

an expression that is valid within the circle considered, then we obtain a
result that is otherwise obvious, by taking

where now g± (— — ) is algebraical and has for its zeros all the zeros within

\Z — d/

the circle ; yu, is an integer; and/j (z) is a function of the same kind as f(z),
which now possesses all the essential singularities of f(z} but has zeros fewer

by the in zeros that are possessed by

z— a

69. Next, consider a function f(z) with n essential singularities al}
a2,..., an but without accidental singularities; and let it have any number of
zeros.

When the zeros are limited in number, they may be taken to be isolated
points, distinct in position from the essential singularities.

When the zeros are unlimited in number, then at least one of the
singularities must be such that an infinite number of the zeros lie within
a circle of finite radius, described round it as centre and containing no other
singularity. For if there be not an infinite number in such a vicinity
of some one point (which can only be an essential singularity, otherwise the
function would be zero everywhere), then the points are isolated and there
must be an infinite number at z = oo . If z = oo be an essential singularity, the
above alternative is satisfied : if not, the function, being continuous save at
singularities, must be zero at all other parts of the plane. Hence it follows
that if a uniform function have a finite number of essential singularities and
an infinite number of zeros, all but a finite number of the zeros lie within
circles of finite radii described round the essential singularities as centres ;
at least one of the circles contains an infinite number of the zeros, and some
of the circles may contain only a finite number of them.

We divide the whole plane into regions, each containing one but only one
singularity and containing also the circle round the singularity ; let the
region containing a{ be denoted by Ci, and let the region Gn be the part of
the plane other than Glt (72, ...... , Gn_^.

If the region G1 contain only a limited number of the zeros, then, by § 68,
we can choose a new function /i (z) such that, if

the function /j (z) has av for an ordinary point, has no zeros within the region
Glt and has a2, a3, ...... , an for its essential singularities.

If the region Cl contain an unlimited number of the zeros, then, as in
Corollaries II. and III. of § 63, we construct any transcendental function

108 GENERAL FORM OF A FUNCTION [69.

5xf— — ) , having a^ for its sole essential singularity and the zeros in GI for

\z — OiJ

all its zeros. When we introduce a function g: (2), defined by the equation

the function g:{z) has no zeros in GI and certainly has a2, a3, ...... , an for

essential singularities ; in the absence of the generalising factor of Glt it can
have Hi for an essential singularity. By § G7, the function ~g{ (2), defined by

gi (z) = 0 - cOc' ehl ^W ,

has no zero and no accidental singularity, and it has a^ as its sole essential
singularity : hence, properly choosing cx and hi, we may take

ft(*)-?i(*)/i(*)«

so that fi (z) does not have aj as an essential singularity, but it has all the
remaining singularities of ^ (z), and it has no zeros within C^.
In either case, we have a new function ft (z) given by

where /^ is an integer, the zeros off(z) that lie in GI are the zeros of GI ; the
function fi(z) has «2, »s> ...... > an (but not a^ for its essential singularities,

and it has the zeros of f(z) in the remaining regions for its zeros.
Similarly, considering (72, we obtain a function /2 (z), such that

where /A.2 is an integer, G2 is a transcendental function finite everywhere except
at a2 and has for its zeros all the zeros of ft (z) — and therefore all the zeros of
f (z) — that lie in G2 ; then f.2 (z) possesses all the zeros of f(z) in the regions
other than GI and C2, and has a3, a4,..., an for its essential singularities.

Proceeding in this manner, we ultimately obtain a function fn (z) which
has none of the zeros off(z) in any of the n regions GI, C2,..., Cn, that is, has
no zeros in the plane, and it has no essential singularities ; it has no acci

dental singularities, and therefore fn(z) is a constant. Hence, when we

•
substitute, and denote by S* (z} the product II (z — a^1, we have

Z —

as the most general form of a function having n essential singularities, no
accidental singularities, and any number of zeros. The function S* (z) is a
rational algebraical function of z, usually meromorphic inform, and it has the
essential singularities off(z) as its zeros and poles ; and the zeros of f (z) are
distributed among the functions Gt.

As however the distribution of the zeros by the regions C and therefore

69.] WITH ESSENTIAL SINGULARITIES 109

the functions G[ ) are somewhat arbitrary, the above form though general

\z — a]

is not unique.

If any one of the singularities, say am, had been accidental and not

essential, then in the corresponding form the function Gm ( - - ) would be

\Z — dm/

algebraical arid not transcendental.

70. A function f(z], which has any finite number of accidental singu
larities in addition to n assigned essential singularities and any number of
assigned zeros, can be constructed as follows.

Let A (z) be the algebraical polynomial which has, for its zeros, the
accidental singularities of f(z), each in its proper multiplicity. Then the
product

/(*)-A(*)

is a function which has no accidental singularities ; its zeros and its essential
singularities are the assigned zeros and the assigned essential singularities of
f (z) and therefore it is included in the form

n (
S*(z)U \0t

i=i (

where S* (z) is a rational algebraical meromorphic function having the points
Oi, a.,,..., an for zeros and poles. The form of the function f(z} is therefore

A

)}•

-ail)

71. A function f (z), which has an unlimited number of accidental singu
larities in addition to n assigned essential singularities and any number of
assigned zeros, can be constructed as follows.

Let the accidental singularities be of, /3',.... Construct a function f^ (z),
having the n essential singularities assigned to f (z}, no accidental singu
larities, and the series a!, /3',. . . of zeros. It will, by § 69, be of the form of a
product of n transcendental functions Gn+1,..., G.2n which are such that a
function G has for its zeros the zeros oif-i(z} lying within a region of the plane,
divided as in § 69 ; and the function Gn+t is associated with the point at-.

Thus / (z) = T*-(z) ft Gn

f=i

where T* (z) is a rational algebraical meromorphic function having its zeros
and its poles, each of finite multiplicity, at the essential singularities ofy(^).
Because the accidental singularities of f(z) are the same points and have
the same multiplicity as the zeros of /i (z), the function / (z) /x (z) has no
accidental singularities. This new function has all the zeros of f(z), and
al,...,an are its essential singularities; moreover, it has no accidental singu
larities. Hence the product f(z)fi (z) can be represented in the form

110 GENERAL FORM OF A FUNCTION [71.

and therefore we have

z-fi

(f-a)

as an expression of the function.

But, as by their distribution through the n selected regions of the plane
in § 69, the zeros can to some extent be arbitrarily associated with the
functions Gl} G*,,..., Gn and likewise the accidental singularities can to some
extent be arbitrarily associated with the functions Gn+l, Gn+»,..., G^i, the
product-expression just obtained, though definite in character, is not unique
in the detailed form of the functions which occur.

S* (z)
The fraction 7**) \

is algebraical and rational ; and it vanishes or becomes infinite only at the
essential singularities alt a.2,..., an, being the product of factors of the form
(z — «i)ms for i = l, 2,..., n. Let the power (z — a^ be absorbed into the
function G{/Gn+i for each of the n values of i ; no substantial change in the
transcendental character of Gi and of Gn+i is thereby caused, and we may
therefore use the same symbol to denote the modified function after the
absorption. Hence "f" the most general product-expression of a uniform
function of z which has n essential singularities al} a*,..., an, any unlimited
number of assigned zeros and any unlimited number of assigned accidental
singularities is

n ^

n —

\z-an

The resolution of a transcendental function with one essential singularity
into its primary factors, each of which gives only a single zero of the function,
has been obtained in § 63, Corollary IV.

We therefore resolve each of the functions G^..., Gm into its primary
factors. Each factor of the first n functions will contain one and only one zero
of the original functions / (.z) ; and each factor of the second n functions will
contain one and only one of the poles of f(z). The sole essential singularity
of each primary factor is one of the essential singularities off(z). Hence we
have a method of constructing a uniform function with any finite number of
essential singularities as a uniformly converging product of any number of
primary factors, each of which has one of the essential singularities as its sole
essential singularity and either (i) has as its sole zero either one of the zeros

t Weierstrass, I.e., p. 48.

71.] WITH ESSENTIAL SINGULARITIES 111

or one of the accidental singularities of/(V), so that it is of the form

Z — € \ a ( — .

or (ii) it has no zero and then it is of the form

/fe).

When all the primary factors of the latter form are combined, they constitute
a generalising factor in exactly the same way as in § 52 and in § 63,
Cor. III., except that now the number of essential singularities is not
limited to unity.

Two forms of expression of a function with a limited number of essential
singularities have been obtained : one (§ 65) as a sum, the other (§ 69) as a
product, of functions each of which has only one essential singularity. Inter
mediate expressions, partly product and partly sum, can be derived, e.g.
expressions of the form

z— c.

But the pure product-expression is the most general, in that it brings into
evidence not merely the n essential singularities but also the zeros and the
accidental singularities, whereas the expression as a sum tacitly requires that
the function shall have no singularities other than the n which are essential.

Note. The formation of the various elements, the aggregate of which is the complete
representation of the function with a limited number of essential singularities, can be
carried out in the same manner as in § 34 ; each element is associated with a particular
domain, the range of the domain is limited by the nearest singularities, and the aggregate
of the singularities forms the boundary of the region of continuity.

To avoid the practical difficulty of the gradual formation of the region of continuity
by the construction of the successive domains when there is a limited number of
singularities (and also, if desirable to be considered, of branch-points), Fuchs devised
a method which simplifies the process. The basis of the method is an appropriate change
of the independent variable. The result of that change is to divide the plane of the
modified variable f into two portions, one of which, G2, is finite in area and the other of
which, Gl, occupies the rest of the plane; and the boundary, common to Gl and G2, is a
circle of finite radius, called the discriminating circle* of the function. In G2 the
modified function is holomorphic ; in G^ the function is holomorphic except at f = oo ;
and all the singularities (and the branch-points, if any) lie on the discriminating circle.

The theory is given in Fuchs's memoir " Ueber die Darstellung der Functionen com-

plexer Variabeln, ," Crelle, t. Ixxv, (1872), pp. 176 — 223. It is corrected in details

and is amplified in Crelle, t. cvi, (1890), pp. 1 — 4, and in Crelle, t. cviii, (1891),
pp. 181—192; see also Nekrassoff, Math. Ann., t. xxxviii, (1891), pp. 82—90, and
Anissimoff, Math. Ann., t. xl, (1892), pp. 145—148.

* Fuchs calls it Grenzkreis.

CHAPTER VII.

FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, AND EXPANSION

IN SERIES OF FUNCTIONS.

72. IT now remains to consider functions which have an infinite number
of essential singularities*. It will, in the first place, be assumed that the
essential singularities are isolated points, that is, that they do not form a
continuous line, however short, and that they do not constitute a continuous
area, however small, in the plane. Since their number is unlimited and
their distance from one another is finite, there must be at least one point in
the plane (it may be at z = oo ) where there is an infinite aggregate of such
points. But no special note need be taken of this fact, for the character of an
essential singularity has not yet entered into question ; the essential singu
larity at such a point would merely be of a nature different from the essential
singularity at some other point.

We take, therefore, an infinite series of quantities a1} a.2, a3,... arranged in
order of increasing moduli, and such that no two are the same : and so we
have infinity as the limit of av when v = <x> .

Let there be an associated series of uniform functions of z such that
for all values of i. the function G'i ( ) , vanishing with , has a{ as its

\Z - Of/ Z — Oi

* The results in the present chapter are founded, except where other particular references are
given, upon the researches of Mittag-Leffler and Weierstrass. The most important investigations
of Mittag-Leffler are contained in a series of short notes, constituting the memoir " Sur la th6orie
des fonctions uniformes d'une variable," Comptes Rendus, t. xciv, (1882), pp. 414, 511, 713, 781,
938, 1040, 1105, 1163, t. xcv, (1882), p. 335 ; and in a memoir " Sur la representation analytique
des fonctions monogenes uniformes," Acta Math., t. iv, (1884), pp. 1 — 79. The investigations of
Weierstrass referred to are contained in his two memoirs " Ueber einen functionentheoretischen
Satz des Herrn G. Mittag-Leffler," (1880), and " Zur Functionenlehre," (1880), both included in
the volume Abhandlungen aus der Functionenlehre, pp. 53 — 66, 67 — 101, 102 — 104. A memoir by
Hermite " Sur quelques points de la theorie des fonctions," Acta Soc. Fenn., t. xii, pp. 67 — 94,
Crelle, t. xci, (1881), pp. 54 — 78 may be consulted with great advantage.

72.]

MITTAG-LEFFLER'S THEOREM

113

sole singularity; the singularity is essential or accidental according as
GI is transcendental or algebraical. These functions can be constructed
by theorems already proved. Then we have the theorem, due to Mittag-
Le frier: — It is always possible to construct a uniform analytical function F (z),
having no singularities other than a1} a«, a,, ... and such that for each deter
minate value of v, the difference F (z)-Gv ( ) is finite for z = av and

\z av/

therefore, in the vicinity of av, is expressible in the form P (z — «„).

73. To prove Mittag-Leffler's theorem, we first form subsidiary functions

Fv (z), derived from the functions G as follows. The function Gv (—- — }

\z — aj

converges everywhere in the plane except at the point «„; hence within a
circle z < av\ it is a monogenic analytic function of z, and can therefore be
expanded in a series of positive powers of z which converges uniformly
within the circle, say

z-a

for values of z such that \z\ < av . If a,, be zero, there is evidently no
expansion.

Let e be a positive quantity less than 1, and let elf e2, e3, ... be arbitrarily
chosen positive decreasing quantities, subject to the single condition that 2e
is a converging series, say of sum A : and let e0 be a positive quantity inter
mediate between 1 and e. Let g be the greatest value of ~ f

z — a,

for

points on or within the circumference \z\ = e0 a,|; then, because the series

00

2 v^z* is a converging series, we have, by § 29,

or

Hence, with values of z satisfying the condition \z\^.e av\, we have, for
any value of m,

/j.=m

Vu Z

2, q -

9 mJt

n = m fco

1-

since e<e0. Take the smallest integral value of m such that

9

F.

114 MITTAG-LEFFLER'S

it will be finite and may be denoted by mv : and thus we have

[73.

for values of z satisfying the condition \z\^.e av\.

We now construct a subsidiary function Fv (z) such that, for all values of z,

then for values of UL which are ^ e aJ,

Moreover, the function 2 zv^ is finite for all finite values of z so that, if we
n=o

take

.j

-a

— i

then 6,,(^) is zero at infinity, because, when 5=00, #„(- -)is finite by

\z — civ/

hypothesis. Evidently <f>v(z) is infinite only at z = av, and its singularity is
of the same kind as that of Gt

z — a,

74. Now let c be any point in the plane, which is not one of the points
«], a2, as, ...; it is possible to choose a positive quantity p such that no one
of the points a is included within the circle

z — c

= p-

Let av be the singularity, which is the point nearest to the origin satisfy
ing the condition «„ > c \ + p ; then, for points within or on the circle, we

have

' z

as

when s has the values v, v + 1, v + 2, Introducing the subsidiary functions

Fv (z), we have, for such values of z,

and therefore

F.(z)

a finite quantity. It therefore follows that the series 2 F, (z) converges uni-

8=v

formly and unconditionally for all values of z which satisfy the condition

74.] THEOREM 115

z — c\^.p. Moreover, all the functions Fl(z), F2(z), ..., Fr_l(z] are finite for
such values of z, because their singularities lie without the circle z — c = p ;
and therefore the series

S Fr(z)

r=l

converges uniformly and unconditionally for all points z within or on the
circle \z — c =p, where p is chosen so that the circle encloses none of the
points a.

The function, represented by the series, can therefore be expanded in the
form P (z — c), in the domain of the point c.

If am denote any one of the points a1} a2, ..., and we take p' so small that
all the points, other than am, lie without the circle

I /

I * U"m — P )

then, since Fm (z) is the only one of the functions F which has a singularity
at am, the series

^{Fr(z}}-Fm(z)

converges regularly in the vicinity of a, and therefore it can be expressed in
the form P (z — am). Hence

a

the difference of Fm and Gm being absorbed into the series P to make Pj. It

GO

thus appears that the series 2 Fr (z) is a function which has infinities only

r = \

at the points a1} a2, ..., and is such that

can be expressed in the vicinity of am in the form P (z - am). Hence 2 Fr (z)
is a function of the required kind.

75. It may be remarked that the function is by no means unique. As
the positive quantities e were subjected to merely the single condition that
they form a converging series, there is the possibility of wide variation in
their choice: and a difference of choice might easily lead to a difference
in the ultimate expression of the function.

This latitude of ultimate expression is not, however, entirely unlimited.
For, suppose there are two functions F(z) and F (z\ enjoying all the assigned
properties. Then as any point c, other than a^, a2, ..., is an ordinary point for
both F (z) and F (z), it is an ordinary point for their difference : and so

F(z)-F(z) = P(z-c)

8—2

116 FUNCTIONS POSSESSING [75.

for points in the immediate vicinity of c. The points a are, however,
singularities for each of the functions : in the vicinity of such a point a*
we have

since the functions are of the required form : hence

F(z}-F(z}=P(z-ai) -P(z- ai),

or the point a; is an ordinary point for the difference of the functions. Hence
every finite point in the plane, whether an ordinary point or a singularity
for each of the functions, is an ordinary point for the difference of the
functions : and therefore that difference is a uniform integral function of z.
It thus appears that, if F (z) be a function with the required properties, then
every other function with those properties is of the form

F(z) + G(z],

where G (z) is a uniform integral function of z either transcendental or
algebraical.

The converse of this theorem is also true.

00

Moreover, the function G (z) can always be expressed in a form 2 gv(z), if

v=\

it be desirable to do so : and therefore it follows that any function with the
assigned characteristics can be expressed in the form

76. The following applications, due to Weierstrass, can be made so as to
give a new expression for functions, already considered in Chapter VI., having
z = oo as their sole essential singularity and an unlimited number of poles at
points Oi, a2, —

If the pole at af be of multiplicity mi} then (z — a$n>f(z) is regular at
the point a; and can therefore be expressed in the form

mi— 1

Hence, if we take /f (z) = 2 c^ (z — ai)~TO<+'t,

M = 0

we have f(z} =fi (z) + P (z — «;).

Now deduce from fi(z) a function Fi(z) as in | 73, and let this deduction be

effected for each of the functions /,- (z). Then we know that

is a uniform function of z having the points a1} a2, ... for poles in the proper

76.] UNLIMITED SINGULARITIES 117

multiplicity and no essential singularity except z = oo . The most general
form of the function therefore is

r=\

Hence any uniform analytical function which has no essential singularity
except at infinity can be expressed as a sum of functions each of which has only
one singularity in the finite part of the plane. The form of Fr (z) is

fr(z}-Gr(z\

where fr (z) is infinite at z = ar and Gr (z) is a properly chosen integral
function.

We pass to the case of a function having a single essential singularity at
c and at no other point and any number of accidental singularities, by taking

z' = - as in § 63. Cor. II.: and so we obtain the theorem :

z — c

Any uniform function which has only one essential singularity, which is
at c, can be expressed as a sum of uniform functions each of which has only
one singularity different from c.

Evidently the typical summative function Fr (z) for the present case is of
the form

Z —

77. The results, which have been obtained for functions possessed of
an infinitude of singularities, are valid on the supposition, stated in § 72,
that the limit of av with indefinite increase of v is infinite ; the series
ttj, «2, ••• tends to one definite limiting point which is 2=00 and, by the
substitution z' (z — c) = 1, can be made any point c in the finite part of the
plane.

Such a series, however, does not necessarily tend to one definite limiting
point: it may, for instance, tend to condensation on a curve, though the
condensation does not imply that all points of the continuous arc of the curve
must be included in the series. We shall not enter into the discussion
of the most general case, but shall consider that case in which the series of
moduli \al) a2 , ... tends to one definite limiting value so that, with in
definite increase of v, the limit of \av is finite and equal to R ; the points
«i, «2, ... tend to condense on the circle \z = R.
Such a series is given by

2fori
( I _ l -m+n

«„,*={! +

for &=0, 1, ..., n, and n=l, 2, ... ad inf.; and another* by

a«Hl + (-l)ncn}e2M7"V2,
where c is a positive proper fraction.

* The first of these examples is given by Mittag-Leffler, Acta Math., t. iv, p. 11 ; the second
was stated to me by Mr Burnside.

118 FUNCTIONS POSSESSING [77.

With each point am we associate the point on the circumference of the
circle, say bm, to which am is nearest: let

| dm "m I = Pm>

so that pm approaches the limit zero with indefinite increase of m. There
cannot be an infinitude of points ap, such that pp^<&, any assigned positive
quantity ; for then either there would be an infinitude of points a within or
on the circle \z\ = R — ®, or there would be an infinitude of points a within
or on the circle z = R + ©, both of which are contrary to the hypothesis
that, with indefinite increase of v, the limit of \av is R. Hence it follows
that a finite integer n exists for every assigned positive quantity ®, such that

\am-bm\ < ®
when m^n.

Then the theorem, which corresponds to Mittag-LefHer's as stated in § 72
and which also is due to him, is as follows : —

It is always possible to construct a uniform analytical function of z which
exists over the whole plane, except at the points a and b, and which, in the
immediate vicinity of each one of the singularities a, can be expressed in the form

where the functions G{ are assigned functions, vanishing with - - and finite

Z — (Li

everywhere in the plane except at the single points a; with which they are
respectively associated.

In establishing this theorem, we shall need a positive quantity e less than
unity and a converging series e^ e2, e3, ... of positive quantities, all less than
unity.

Let the expression of the function Gn be

"I / _. .. \0 I / - _. \5 ' ' ' ' '

n \z - a ~ z-an (z- an)2 (z - an)s

Then, since z - an = (z - bn) \l -- — ~l\ ,

( z on )

the function Gn can be expressed* in the form

l«— <li

for values of z such that

an -

z-bn
and the coefficients A are given by the equations

* The justification of this statement is to be found in the proposition in § 82.

77.]

UNLIMITED SINGULARITIES

119

Now, because Gn is finite everywhere in the plane except at an, the series

has a finite value, say #, for any non-zero value of the positive quantity %n ;
then

Hence

0*-!)!

ft & f

< S flfr-^

71 ?^

Introducing a positive quantity a such that

we choose £n so that £n < a|an - bn\ ;

and then | A n> ^ \ < go. ( 1 + a)*-1.

Because (1 + a) e is less than unity, a quantity 6 exists such that

(1 + a) e < 6 < 1.

Then for values of z determined by the condition

go. 6

dn on

< e, we have

al-0'

Let the integer mn be chosen so that

ga &>

it will be a finite integer, because 0< 1. Then

00 (1 7)

V I A I "H ^^

We now construct, as in § 73, a subsidiary function Fn(z), defining it by
the equation

so that for points z determined by the condition

\Fn(z)\<en.
A function with the required properties is

00

Fm(z\

< €, we have

m=l

120 FUNCTIONS POSSESSING [77.

To prove it, let c be any point in the plane distinct from any of the points
a and b ; we can always find a value of p such that the circle

\z-c\=p

contains none of the points a and b. Let I be the shortest distance between
this circle and the circle of radius R, on which all the points b lie ; then for

all points z within or on the circle

z — c

— p we have

Now we have seen that, for any assigned positive quantity <s), there is a
finite integer n such that

I dm — bm < ©

when m ^ n. Taking ® = el, we have

m

< e

when m^n,n being the finite integer associated with the positive quantity el.
It therefore follows that, for points z within or on the circle \z — c\ = p,

\Fm(z}\<em,
when m is not less than the finite integer n. Hence

a finite quantity because e1} e2, ... is a converging series; and therefore

is a converging series. Each of the functions F1(z), F»(z), ..., Fn_-i(z) is
finite when z — c ^ p ; and therefore

is a series which converges uniformly and unconditionally for all values of z
included in the region

\z-c\^p.

Hence the function represented by the series can be expressed in the form
P (z — c) for all such values of z. The function therefore exists over the
whole plane except at the points a and b.

It may be proved, exactly as in § 74, that, for points z in the immediate
vicinity of a singularity am,

The theorem is thus completely established.

The function thus obtained is not unique, for a wide variation of choice of
the converging series ea + e2 + . . . is possible. But, in the same way as in the

77.]

UNLIMITED SINGULARITIES

121

corresponding case in § 75, it is proved that, if F (z) be a function with the
required properties, every other function with those properties is of the form

F(z}+G(z\

where G (z) behaves regularly in the immediate vicinity of every point in the
plane except the points b.

78. The theorem just given regards the function in the light of an
infinite converging series of functions of the variable : it is natural to suppose
that a corresponding theorem holds when the function is expressed as an
infinite converging product. With the same series of singularities as in
§ 77, when the limit of av with indefinite increase of v is finite and
equal to R, the theorem* is: —

It is always possible to construct a uniform analytical function which
behaves regularly everywhere in the plane except at the points a and b and
which in the vicinity of any one of the points av can be expressed in the
form

where the numbers w1} n2, ... are any assigned integers.

The proof is similar in details to proofs of other propositions and it will
therefore be given only in outline. We have

au-

provided

such values of z,

z-av z-bv z - bv ^i V z - bv J '
< e, the notation being the same as in § 77. Hence, for

=e

(/7 _ 7) \
i_ ^ _ M
2-bJ

-n,, S

by Ev (z), we have Ev (z} =e m"

Hence, if F(z) denote the infinite product

we have F(z) = e

and F(z) is a determinate function provided the double series in the index of

the exponential converge.

* Mittag-Leffler, Acta Math., t. iv, p. 32 ; it may be compared with Weierstrass's theorem in
§67.

122 TRANSCENDENTAL FUNCTION AS

Because nv is a finite integer and because

[78.

is a converging series, it is possible to choose an integer mv so that

7)

"x

M(T^

where t]v is any assigned positive quantity. We take a converging series of
positive quantities rjv : and then the moduli of the terms in the double series
form a converging series. The double series itself therefore converges
uniformly and unconditionally ; and then the infinite product F (z) converges
uniformly and unconditionally for points z such that

&„ — b..

< e.

As in § 77, let c be any point in the plane, distinct from any of the
points a and b. We take a finite value of p such that the circle z — c\=p
contains none of the points a and b ; and then, for all points within or on this
circle,

z—

<e

when m^n, n being the finite integer associated with the positive quantity
el. The product

fi Ev(z)

v=n

is therefore finite, for its modulus is less than

CO

S IJK

K = »

the product

n

v=l

is finite, because the circle z — c\ = p contains none of the points a and 6;
and therefore the function F(z) is finite for all points within or on the circle.
Hence in the vicinity of c, the function can be expanded in the form P (z — c) ;
and therefore the function exists everywhere in the plane except at the points
a and b.

The infinite product converges ; it can be zero only at points which make
one of the factors zero and, from the form of the factors, this can take place
only at the points av with positive integers nv. In the vicinity of av all
the factors of F (z) except Ev (z) are regular ; hence F (z)\Ev (z) can be
expressed as a function of z — av in the vicinity. But the function has no
zeros there, and therefore the form of the function is

Pl (z-a,,).

78.] AN INFINITE SERIES OF FUNCTIONS 123

Hence in the vicinity of av, we have

on combining with Pl (z — av) the exponential index in Ev(z). This is the
required property.

Other general theorems will be found in Mittag-Leffler's memoir just
quoted.

79. The investigations in §§ 72 — 75 have led to the construction of a
function with assigned properties. It is important to be able to change, into
the chosen form, the expression of a given function, having an infinite series
of singularities tending to a definite limiting point, say to z = oo . It is
necessary for this purpose to determine (i) the functions Fr(z) so that the

00

series 2 Fr (z) may converge uniformly and (ii) the function G (z).

r=l

Let <& (z) be the given function, and let S be a simple contour embracing
the origin and /j, of the singularities, viz., al , ...... , aM: then, if t be any

point, we have

- « « .

m r« *£) ,,y r« *y) ,,.

J t-z\t) J t-z\t)

f(a) _

where I implies an integral taken round a very small circle centre a.

If the origin be one of the points a1} a2, ...... , then the first term will be

included in the summation.

Assuming that z is neither the origin nor any one of the points a1} ..., a^,
we have

so

27TI

AT ^

Now — . 7-^-7 dt

1 [(0)$>(t)fz\
— . 7-^-7
Ziri] t-z\tj

,
— -—. 2 I 7—^- - dt.

t-Z\t)

(ffl-l)i I

\~dm 1(®(t) + ^i^+^

[

124 TRANSCENDENTAL FUNCTION AS [79.

unless z = 0 be a singularity and then there will be no term G (z). Similarly,
it can be shewn that

/ I \ m-l / z \ A

is equal to Gv(- -} - 2 vj-} = F, (z),

\z - aj A=0 \aj

where , — s— •

2?rt

and the subtractive sum of m terms is the sum of the first m terms in the
development of Gv in ascending powers of z. Hence

If, for an infinitely large contour, m can be chosen so that the integral

t-

diminishes indefinitely with increasing contours enclosing successive singu
larities, then

The integer m may be called the critical integer.
If the origin be a singularity, we take

and there is then no term G (z) : hence, including the origin in the summa
tion, we then have

so that if, for this case also, there be some finite value of m which makes
the integral vanish, then

Other expressions can be obtained by choosing for m a value greater than
the critical integer ; but it is usually most advantageous to take m equal to
its least lawful value.

Ex. 1. The singularities of the function ?r cot 772 are given by z = \, for all integer
values of X from — oo to +00 including zero, so that the origin is a singularity.

The integral to be considered is

- 1 M IT cot vt fz\m ,,
= ~ — . I — - (- ) at.

2iri J t-z \tj

We take the contour to be a circle of very large radius R chosen so that the circumference
does not pass innuitesimally near any one of the singularities of TT coint at infinity; this

79.] AN INFINITE SERIES OF FUNCTIONS 125

is, of course, possible because there is a finite distance between any two of them. Then,
round the circumference so taken, n cot nt is never infinite : hence its modulus is never
greater than some finite quantity M.

Let t = Reei, so that ~=id6; then

v

and therefore

Z

.--—.
t-z

for some point t on the circle. Now, as the circle is very large, we have \t-z\ infinite :
hence \J\ can be made zero merely by taking m unity.

Thus, for the function TT cot TTZ, the critical integer is unity.
Hence from the general theorem we have the equation

1 fir cot nt z j

7T COt 772= -5— . 2 I— -dt,

2TTI J t-Z t

the summation extending to all the points X for integer values of X = - oc to + oo , and
each integral being taken round a small circle centre X.

-vr . » . 1 /"(*) TT cot irt z ,

Now if, in -— . • -dt.

2m J t - z t

we take t=\ + (, we have

where P(Q = 0 when £= 0; and therefore the value of the integral is

•*./ (*-*+{) (x+fl t

In the limit when |f| is infinitesimal, this integral

z

= (X-2)X

1 1

~X-2 X'

and therefore /*. (z) = -J— + 1

A ' z-X X'

if X be not zero.

And for the zero of X, the value of the integral is

(p

126 REGION OF CONTINUITY [79.

so that F0(z) is -. In fact, in the notation of § 72, we have

z

o P-A»JL

^ \z-\J~z-\'
arid the expansion of GK needs to be carried only to one term.

1 A=ao /I 1\
We thus have 7rcot7rs = — f- 2 — N+=r)>

z A=-co \Z-X A/

the summation not including the zero value of X.
Ex. 2. Obtain, ab initio, the relation

SHI2 3 A=_aj (z-X7r)2'

p. 3. Shew that, if

1 °° 1 1

then "-^^ = - + 2z 2 ^3-^1-

R(z) z i=lR(\)z*-\*

(Gylden, Mittag-Leffler.)

Ex. 4. Obtain an expression, in the form of a sum, for

IT cot irz

where Q(z) denotes (1 -z) (l -^ (l -|J ...... ^-j)*-

80. The results obtained in the present chapter relating to functions
which have an unlimited number of singularities, whether distributed over
the whole plane or distributed over only a finite portion of it, shew that
analytical functions can be represented, not merely as infinite converging
series of powers of the variable, but also as infinite converging series of
functions of the variable. The properties of functions when represented by
series of powers of the variable depended in their proof on the condition that
the series proceeded in powers; and it is therefore necessary at least to
revise those properties in the case of functions when represented as series
of functions of the variable.

Let there be a series of uniform functions /i (z), /, (z), . . . ; then the
aggregate of values of z, for which the series

1*1

has a finite value, is the region of continuity of the series. If a positive
quantity p can be determined such that, for all points z within the circle

z — a\ = p,

80.] OF A SERIES OF FUNCTIONS 127

00

the series 2 fi(z) converges uniformly and unconditionally*, the series is

said to converge in the vicinity of a. If R be the greatest value of p for
which this holds, then the area within the circle

z — a\ = R

is called the domain of a; and the series converges uniformly and uncon
ditionally in the vicinity of any point in the domain of a.

It will be proved in § 82 that the function can be represented by power-
series, each such series being equivalent to the function within the domain of
some one point. In order to be able to obtain all the power-series, it is
necessary to distribute the region of continuity of the function into domains
of points where it has a uniform, finite value. We therefore form the domain
of a point 6 in the domain of a from a knowledge of the singularities of the
function, then the domain of a point c in the domain of 6, and so on ; the
aggregate of these domains is a continuous part of the plane which has
isolated points and which has one or several lines for its boundaries. Let
this part be denoted by At.

For most of the functions, which have already been considered, the region
A1} thus obtained, is the complete region of continuity. But examples will
be adduced almost immediately to shew that A-^ does not necessarily include
all the region of continuity of the series under consideration. Let a' be a
point not in A-^ within whose vicinity the function has a uniform, finite
value ; then a second portion A2 can be separated from the whole plane, by
proceeding from a' as before from a. The limits of A± and A2 may be wholly
or partially the same, or may be independent of one another : but no point
within either can belong to the other. If there be points in the region of con
tinuity which belong to neither A1 nor A2, then there must be at least another
part of the plane A3 with properties similar to At and^l2- And so on. The

00

series 2 fi(z) converges uniformly and unconditionally in the vicinity of

»=i «

every point in each of the separate portions of its region of continuity.

It was proved that a function represented by a series of powers has a
definite finite derivative at every point lying actually within the circle
of convergence of the series, but that this result cannot be affirmed for a
point on the boundary of the circle of convergence even though the value of
the series itself should be finite at the point, an illustration being provided
by the hypergeometric series at a point on the circumference of its circle of

* In connection with most of the investigations in the remainder of this chapter, Weierstrass's
memoir " Zur Functionenlehre " already quoted (p. 112, note) should be consulted.

It may be convenient to give here Weierstrass's definition (I.e., p. 70) of uniform, unconditional

convergence. A series 2 fn converges uniformly, if an integer m can be determined so that

/»

can be made less than any arbitrary positive quantity, however small ; and it converges uncon
ditionally, if the uniform convergence of the series be independent of any special arrangement
of order or combination of the terms.

128 REGION OF CONTINUITY OF [80.

convergence. It will appear that a function represented by a series of
functions has a definite finite derivative at every point lying actually within
its region of continuity, but that the result cannot be affirmed for a point
on the boundary; and an example will be given (§ 83) in which the derivative
is indefinite.

Again, it has been seen that a function, initially defined by a given power-
series, is, in most cases, represented by different analytical expressions in
different parts of the plane, each of the elements being a valid expression of
the function within a certain region. The questions arise whether a given
analytical expression, either a series of powers or a series of functions :
(i) can represent different functions in the same continuous part of its region
of continuity, (ii) can represent different functions in distinct (that is, non-
continuous) parts of its region of continuity.

81. Consider first a function defined by a given series of powers.

Let there be a region A' in the plane and let the region of continuity of
the function, say g (z), have parts common with A'. Then if a0 be any point
in one of these common parts, we can express g {z) in the form P (z — a0) in
the domain of a0.

As already explained, the function can be continued from the domain of
a0 by a series of elements, so that the whole region of continuity is gradually
covered by domains of successive points ; to find the value in the domain of
any point a, it is sufficient to know any one element, say, the element in the
domain of a0. The function is the same through its region of continuity.

Two distinct cases may occur in the continuations.

First, it may happen that the region of continuity of the function g (z)
extends beyond A'. Then we can obtain elements for points outside A',
their aggregate being a uniform analytical function. The aggregate of
elements then represents within A' a single analytical function : but as that
function has elements for points without A, the aggregate within A' does
not completely represent the function. Hence

If a function be defined within a continuous region of a plane by an
aggregate of elements in the form of power-series, which are continuations of
one another, the aggregate represents in that part of the plane one (and only
one) analytical function : but if the power-series can be continued beyond the
boundary of the region, the aggregate of elements within the region is not the
complete representation of the analytical function.

This is the more common case, so that examples need not be given.

Secondly, it may happen that the region of continuity of the function does
not extend beyond A' in any direction. There are then no elements of the
function for points outside A' and the function cannot be continued beyond
the boundary of A. The aggregate of elements is then the complete
representation of the function and therefore :

81.] A SERIES OF POWERS 129

If a function be defined within a continuous region of a plane by an
aggregate of elements in the form of power-series, which are continuations of
one another, and if the power-series cannot be continued across the boundary of
that region, the aggregate of elements in the region is the complete representa
tion of a single uniform monogenic function which exists only for values of the
variable within the region.

The boundary of the region of continuity of the function is, in the latter
case, called the natural limit of the function*, as it is a line beyond which
the function cannot be continued. Such a line arises for the series

l + 2z + ^ + 2z9 + ... ,

in the circle \z = 1, a remark due to Kronecker; other illustrations occur in
connection with the modular functions, the axis of real variables being the
natural limit, and in connection with the automorphic functions (see Chapter
XXII.) when the fundamental circle is the natural limit. A few examples
will be given at the end of the present Chapter.

It appears that Weierstrass was the first to announce the existence of natural limits
for analytic functions, Berlin Monatsber. (1866), p. 617 ; see also Schwarz, Ges. Werke,
t. ii, pp. 240 — 242, who adduces other illustrations and gives some references ; Klein and
Fricke, Vorl. uber die Theorie der elliptischen Modulfunctioncn, t. i, (1890), p. 110; Jordan,
Cows d' Analyse, t. iii, pp. 609, 610. Some interesting examples and discussions of
functions, which have the axis of real variables for a natural limit, are given by Hankel,
" Untersuchungen liber die unendlich oft oscillirenden und unstetigen Functionen,"
Math. Ann., t. xx, (1870), pp. 63—112.

82. Consider next a series of functions of the variable ; let it be

The region of continuity may be supposed to consist of several distinct parts,
in the most general case ; let one of them be denoted by A. Take some
point in A, say the origin, which is either an ordinary point or an isolated
singularity; and let two concentric circles of radii R and R' be drawn in A,
so that

R < z =r<R,

and the space between these circles lies within A. In this space, each term
of the series is finite and the whole series converges uniformly and uncon
ditionally.

Now let fi (z) be expanded in a series of powers of z, which series con
verges within the space assigned, and in that expansion let ^ be the co-

oo

efficient of z* ; then we can prove that 2 i^ is finite and that the series

( / °° \

s |(sO

n. (\i = 0 I

* Die natiirliche Grenze, according to German mathematicians.
F.

130 REGION OF CONTINUITY [82.

converges uniformly and unconditionally within this space, so that

•x. (/ oo

2 /,(*) = 2 2

i=l " /A {\i=Q

00

Because the infinite series 2 fi (z) converges uniformly and uncon
ditionally, a number n can be chosen so that

where & is an arbitrary finite quantity, ultimately made infinitesimal; and
therefore also

i=n

where n' > n and is infinite in the limit. Now since the number of terms in
the series

is not infinite before the limit, we have

But the original series converges unconditionally, and therefore k is not less

n

than the greatest value of the modulus of 2 fi(z) for points within the

i=n

region; hence, by § 29, we have

n

2 V < AT <i.
»•=»

00

Moreover, A; is not less than the greatest value of the modulus of 2 fi(z)
in the given region ; and so

00

2 i^ < AT *.

i=n

Now, by definition, k can be made as small as we desire by choice of n ; hence

the series

is a converging series. Let it be denoted by A^.

n-l oo

Let 2 r'M = A /, 2 ifj, = A M" ;

then, by the above suppositions, we can always choose n so that

k being any assignable small quantity.

82.]

OF A SERIES OF FUNCTIONS

131

When two new quantities r± and r2 are introduced, as in § 28, satisfying
the inequalities

f-f ^ ly ^ \ iv --• /y» ^ 7?'

-il/<^/l<s.|.S|<i./2<.-fl,

the integer w can be chosen so that

\Ap'\ < kr~* < kr^.

f- r.

Then

and

so that

2

.—

00

2

-

- <k

M=-oo r — r-i r2-r

Hence the series 2 A^'z^ can by choice of n be made to have a
modulus less than any finite quantity ; and therefore, since

/u.= oo n — 1

(for there is a finite number of terms in the coefficients on each side, the
expansions are converging series, and the sum on the right-hand side is a
finite quantity), it follows that the series

converges uniformly.
Finally, we have

2 .

fl= —00

2 ft (*) - 24^ = 2 /< (z) -
<=i 1=1

and therefore

2

t'=n

r ~

which, as k can be diminished indefinitely, can be made less than any finite

jlX=00

quantity. Hence the series 2 A^ converges unconditionally, and there-

fi= —00

fore we have

provided

00 jlt=00

2 /;(*)= 2 .

l'=l /u= — oo

9—2

132 REGION OF CONTINUITY [82.

When we take into account all the parts of the region of continuity
of the series, constituted by the sum of the functions, we have similar
expansions in the form of successive series of powers of z — c, converging
uniformly and unconditionally in the vicinities of the successive points c.
But, in forming the domains of these points c, the boundary of the region of
continuity of the function must not be crossed ; and a new series of powers is
required when the circle of convergence of any one series (lying within the
region of continuity) is crossed.

It therefore appears that a converging series of functions of a variable
can be expressed in the form of series of powers of the variable which
converge within the parts of the plane where the series of functions
converges uniformly and unconditionally ; but the equivalence of the two
expressions is limited to such parts of the plane and cannot be extended
beyond the boundary of the region of continuity of the series of functions.

If the region of continuity of a series of functions consist of several parts
of the plane, then the series of functions can in each part be expressed in
the form of a set of converging series of powers : but the sets of series of
powers are not necessarily the same for the different parts, and they are not
necessarily continuations of one another, regarded as power-series.

Suppose, then, that the region of continuity of a series of functions

F(z)=lfi(z)

i=l

consists of several parts A1} A.2, Within the part A^ let F (z) be

represented, as above, by a set of power-series. At every point within A1}
the values of F(z) and of its derivatives are each definite and unique ; so
that, at every point which lies in the regions of convergence of two of the
power-series, the values which the two power-series, as the equivalents of F (z)
in their respective regions, furnish for F (z) and for its derivatives must be
the same. Hence the various power-series, which are the equivalents of F (z)
in the region Aly are continuations of one another: and they are sufficient to
determine a uniform monogenic analytic function, say F^ (z}. The functions
F(z) and Fl(z) are equivalent in the region Al; and therefore, by § 81, the
series of functions represents one and the same function for all points within
one continuous part of its region of continuity. It may (and frequently does)
happen that the region of continuity of the analytical function F± (z) extends
beyond A± ; and then F-^ (z) can be continued beyond the boundary of A^ by
a succession of elements. Or it may happen that the region of continuity
of Fl (z) is completely bounded by the boundary of A^ ; and then the function
cannot be continued across that boundary. In either case, the equivalence

00

of F-L(Z) and 2 fi(z) does not extend beyond the boundary of Alt one

82.] OF A SERIES OF FUNCTIONS 133

00

complete and distinct part of the region of continuity of 2 fi(z); and

i = \

therefore, by using the theorem proved in § 81, it follows that :

A series of functions of a variable, which converges within a continuous part
of the plane of the variable z, is either a partial or a complete representation
of a single uniform, analytic function of the variable in that part of the plane.

83. Further, it has just been proved that the converging series of
functions can, in any of the regions A, be changed into an equivalent
uniform, analytic function, the equivalence being valid for all points in
that region, say

2 /(•). 4(4

i = l

But for any point within A, the function Fl(z) has a uniform finite derivative

oo

(§ 21); and therefore also 2 fi(z) has a uniform finite derivative. The

i=l

equivalence of the analytic function and the series of functions has not been
proved for points on the boundary; even if they are equivalent there, the
function I\ (z) cannot be proved to have a uniform finite derivative at every

00

point on the boundary of A, and therefore it cannot be affirmed that 2 ft (z)

i=\

has, of necessity, a uniform, finite derivative at points on the boundary of A, even

oo

though the value of 2 fi(z) be uniform and finite at every point on the

i=l

boundary*.

Ex. In illustration of the inference just obtained, regarding the derivative of a
function at a point on the boundary of its region of continuity, consider the series

g(z)= 2 &V",

n=0

where b is a positive quantity less than unity, and a is a positive quantity which will be
taken to be an odd integer.

For points within and on the circumference of the circle \z =1, the series converges
uniformly and unconditionally; and for all points without the circle the series diverges.
It thus defines a function for points within the circle and on the circumference, but not
for points without the circle.

Moreover for points actually within the circle the function has a first derivative and
consequently has any number of derivatives. But it cannot be declared to have a
derivative for points on the circle: and it will in fact now be proved that, if a certain
condition be satisfied, the derivative for variations at any point on the circle is not merely
infinite but that the sign of the infinite value depends upon the direction of the variation,
so that the function is not monogenic for the circumference t.

* It should be remarked here, as at the end of § 21, that the result in itself does not contravene
Biemann's definition of a function, according to which (§ 8) -^ must have the same value what
ever be tbe direction of the vanishing quantity dz ; at a point on the boundary of the region
there are outward directions for which die is not defined.

t The following investigation is due to Weierstrass, who communicated it to Du Bois-Eeymond :
see Crclle, t. Ixxix, (1875), pp. 29—31.

134

A SERIES OF FUNCTIONS

[83.

Let z = eei: then, as the function converges unconditionally for all points along the
circle, we take

f(ff)= 2 lnea"ei,

71=0

where 6 is a real variable. Hence

m-l IV,an(0 + 4>)*_,,«WWl

= s«nH - — -
H=O 1 an$ J

/•ea">+»> (0 + <f>) i _ ea™+«0(S

+ 2 &w + M - -T - 1 1

«=o I 9 J

assuming m, in the first place, to be any positive integer. To transform the first sum on
the right-hand side, we take

and therefore

pan (0 + <j>) i _ a"0i

2 (ab}n

n=0

<M21^n 8Jn(fr-*)

if ab>\. Hence, on this hypothesis, we have

2 (ab)n \ — \ =y r i »

*=o ( a"0 J ao - 1

where 7 is a complex quantity with modulus <1.

To transform the second sum on the right-hand side, let the integer nearest to am
be am, so that

7T

for any value of m : then taking

we have \tr^-x> — %n,

and cos x is not negative. We choose the quantity <f> so that

and therefore

TT am

ff) — — — ,
0

which, by taking m sufficiently large (a is > 1), can be made as small as we please. We
now have

am+"(6 +<i>)i = Qaniti (1 + o™) _ _ / _ j N°™

if a be an odd integer, and

_

am+nOi _ ani (x + iram] _ / _ j \<»meana;i
, a"xi

Hence

CD /•

and therefore 2 &- + « f

,,=0 i

_

- ( - 1)

2 6"

83.] MAY NOT POSSESS A DERIVATIVE 135

The real part of the series on the right-hand side is

2 bn{l + cosanx};

n=0

every term of this is positive and therefore, as the first term is 1 + cos x, the real part

> 1+cos.r

>1
for cos x is not negative ; and it is finite, for it is

<2 2 bn

K=0

2

<r^6-

Moreover far < TT — x < frr,

so that -- is positive and >-. Hence
TT — x 6

where TJ is a finite complex quantity, the real part of which is positive and greater than
unity. We thus have

where |y'|<l, and the real part of 77 is positive and > 1.
Proceeding in the same way and taking

IT ' am '

TT+X

so that % = — — ,

we find — — — — t_LJ — _ ( _ iy™ (a^

where |y/|<l and the real part of TJ^ a finite complex quantity, is positive and greater
than unity.

If now we take ab - 1 > fn-,

the real parts of - — + y -*-—= , say of f,

O 7T (tO — 1

and of |li+yi'__L_,sayof fl,

are both positive and different from zero. Then, since

and ~x- = (_!)«- (ab)m d ,

/(.

m being at present any positive integer, we have the right-hand sides essentially different
quantities, because the real part of the first is of sign opposite to the real part of the second.

Now let m be indefinitely increased; then $ and x are infinitesimal quantities
which ultimately vanish ; and the limit of - [/(# + </>)-/(#)] for $ = 0 is a complex infinite

136 ANALYTICAL EXPRESSION [83.

quantity with its real part opposite in sign to the real part of the complex infinite quantity
which is the limit of $ — ~^ f°r = ®- If# had a differential coefficient

A

these two limits would be equal : hence / (0) has not, for any value of 6, a determinate
differential coefficient.

From this result, a remarkable result relating to real functions may be at once derived.
The real part of / (<9) is

2 6ncos(an<9),

n=0

which is a series converging uniformly and unconditionally. The real parts of

-(-ir («&)-<:

and of +(-l)am(a6)TOf1

are the corresponding magnitudes for the series of real quantities : and they are of opposite
signs. Hence for no value of 6 has the series

2 6"cos(an<9)

n=0

a determinate differential coefficient, that is, we can choose an increase <£ and a decrease ^
of 6, both being made as small as we please and ultimately zero, such that the limits of
the expressions

0 -X

are different from one another, provided a be an odd integer and ab > 1 +|TT.

The chief interest of the above investigation lies in its application to functions of real
variables, continuity in the value of which is thus shewn not necessarily to imply the
existence of a determinate differential coefficient defined in the ordinary way. The
application is due to Weierstrass, as has already been stated. Further discussions will
be found in a paper by Wiener, Crelle, t. xc, (1881), pp. 221 — 252, in a remark by
Weierstrass, Abh. aus der Functionenlehre, (1886), p. 100, and in a paper by Lerch, Crelle,
t. ciii, (1888), pp. 126 — 138, who constructs other examples of continuous functions of
real variables ; and an example of a continuous function without a derivative is given by
Schwarz, Ges. Werke, t. ii, pp. 269 — 274.

The simplest classes of ordinary functions are characterised by the properties : —
(i) Within some region of the plane of the variable they are uniform, finite and

continuous :
(ii) At all points within that region (but not necessarily on its boundary) they have

a differential coefficient :

(iii) When the variable is real, the number of maximum values and the number of
minimum values within any given range is finite.

The function 2 bn cos (anQ\ suggested by Weierstrass, possesses the first but not the

71=0

second of these properties. Kb'pcke (Math. Ann., t. xxix, pp. 123 — 140) gives an example
of a function which possesses the first and the second but not the third of these
properties.

84. In each of the distinct portions Alt A.2>... of the complete region of
continuity of a series of functions, the series can be represented by a
monogenic analytic function, the elements of which are converging power-
series. But the equivalence of the function -series and the monogenic

84.] REPRESENTING DIFFERENT FUNCTIONS 137

analytic function for any portion A^ is limited to that region. When the
monogenic analytic function can be continued from A^ into Az, the continua
tion is not necessarily the same as the monogenic analytic function which is

00

the equivalent of the series 2 fi(z) in A2. Hence, if the monogenic analytic

i = l

functions for the two portions A^ and A2 be different, the function-series
represents different functions in the distinct parts of its region of continuity.

A simple example will be an effective indication of the actual existence
of such variety of representation in particular cases ; that, which follows, is
due to Tannery*.

Let a, b, c be any three constants ; then the fraction

a + bczm
Y+'bzm '
when m is infinite, is equal to a if z \ < 1, and is equal to c if | z > 1.

Let m0, m1} m2>... be any set of positive integers arranged in ascending
order and be such that the limit of mn, when n = oo , is infinite. Then,

since

a + bczm* a + bczm° » {a + bczmi a + bczm

1 + bzm» 1 -f bzm° f.i (1 + bzmi I + bz'"
^mo

" ~* a)

the function <f)(z), defined by the equation

,. a + bczm° ., N S f 0^-^-1-1)^-1

+ (z} = TT6^ + b (G ~ a) £ {(I + bz^) (i + 6^

converges uniformly and unconditionally to a value a if \ z < 1, awe? converges
uniformly and unconditionally to a value c if z \ > 1. But it does not con
verge uniformly and unconditionally if z \ = 1.

The simplest case occurs when b = — 1 and m^ = 2* ; then, denoting the
function by <f> (z), we have

a - cz , . ( z z2 z4

that is, the function <f> (z) is equal to a if z < 1, and it is equal to c if

* It is contained in a letter of Tannery's to Weierstrass, who communicated it to the Berlin
Academy in 1881, Abh. aus der Functionenlehre, pp. 103, 104. A similar series, which indeed is
equivalent to the special form of $ (z), was given by Schroder, Schlfim. Zeitschrift, t. xxii, (1876),
p. 184; and Pringsheim, Math. Ann., t. xxii, (1883), p. 110, remarks that it can be deduced,
without material modifications, from an expression given by Seidel, Crelle, t. Ixxiii, (1871),
pp. 297- -299.

138 LINE OF SINGULARITIES [84.

When \z =\, the function can have any value whatever. Hence a circle
of radius unity is a line of singularities, that is, it is a line of discontinuity
for the series. The circle evidently has the property of dividing the plane
into two parts such that the analytical expression represents different
functions in the two parts.

If we introduce a new variable £ connected with z by the relation*

l +z

then, if £= £ + iy and z = x + iy, we have

1 rfS. nil

fc i — x y

so that £ is positive when \z\< 1, and £ is negative when \ z \ > 1. If then

the function %(£) is equal to a or to c according as the real part of f is
positive or negative.

And, generally, if we take £ a rational function of z and denote the
modified form of </> (£), which will be a sum of rational functions of z, by
^(z), then <f>i(z) will be equal to a in some parts of the plane and to c
in other parts of the plane. The boundaries between these parts are lines
of singular points : and they are constituted by the ^-curves which correspond
to £| = 1.

85. Now let F(z) and G(z) be two functions of z with any number of
singularities in the plane : it is possible to construct a function which shall
be equal to F (z} within a circle centre the origin and to G (z) without the
circle, the circumference being a line of singularities. For, when we make
a = 1 and c = 0 in </> (z) of § 84, the function

1 z z* z4

00)=- -- + -. — r + •-: — : + -. -^r + . . .
V/ 1—0 Z2 — I Z*— I ZS —I

is unity for all points within the circle and is zero for all points without it :

and therefore

G(z} + {F(z)-G(z)}6(z}

is a function which has the required property.

Similarly F3 (z) + {F, (z) - F, (z)} 6 (z) + {F, (z) - F3 (z}} 6 (

is a function which has the value Fl (z) within a circle of radius unity, the value F2 (z)
between a circle of radius unity and a concentric circle of radius r greater than unity, and
the value F3(z) without the latter circle. All the singularities of the functions F1} F2, F3
are singularities of the function thus represented; and it has, in addition to these, the
two lines of singularities given by the circles.

* The significance of a relation of this form will be discussed in Chapter XIX.

85.] MONOGENIC FUNCTIONALITY 139

Again, 6

is a function of s, which is equal to F(z) on the positive side of the axis of y, and is equal
to G (z) on the negative side of that axis.

1+2

Also, if we take £e l —p\ = ^~i

where ax and p1 are real constants, as an equation defining a new variable £ + iy, we have
| cos at + 77 sin aj -pl = p. \23T~2

so that the two regions of the 2-plane determined by \z\<l and \z\>l correspond to the
two regions of the {"-plane into which the line £ cos a: + 77 sin al—p1 = 0 divides it. Let

,-«'ai — », — 1\

so that on the positive side of the line £ cos at + 77 sin aj — p1 = 0 the function 6l is unity and
on the negative side of that line it is zero. Take any three lines defined by ax, p1; a2, p2',
a,, pn respectively ; then

AJ.A11 (2)\-F/(l)

is a function which has the value F within
the triangle, the value - F in three of the
spaces without it, and the value zero in the
remaining three spaces without it, as indi
cated in the figure (fig. 13).

And for every division of the plane by
lines, into which a circle can be transformed (3)
by rational equations, as will be explained
when conformal representation is discussed (1) /

hereafter, there is a possibility of represent- Fig. 13.

ing discontinuous functions, by expressions similar to those just given.

These examples are sufficient to lead to the following result*, which is
complementary to the theorem of § 82 :

When the region of continuity of an infinite series of functions consists
of several distinct parts, the series represents a single function in each part
but it does not necessarily represent the same function in different parts.

It thus appears that an analytical expression of given form, which con
verges uniformly and unconditionally in different parts of the plane separated
from one another, can represent different functions of the variable in those
different parts ; and hence the idea of monogenic functionality of a complex
variable is not coextensive with the idea of functional dependence expressible
through arithmetical operations, a distinction first established by Weierstrass.

86. We have seen that an analytic function has not a definite value at
an essential singularity and that, therefore, every essential singularity is
excluded from the region of definition of the function.

* Weierstrass, I.e., p. 90.

140 SINGULAR LINES [86.

Again, it has appeared that not merely must single points be on occasion
excluded from the region of definition but also that functions exist with
continuous lines of essential singularities which must therefore be excluded.
One method for the construction of such functions has just been indicated :
but it is possible to obtain other analytical expressions for functions which
possess what may be called a singular line. Thus let a function have a
circle of radius c as a line of essential singularity*; let it have no other
singularities in the plane and let its zeros be al} a2, a3,..., supposed arranged
in such order that, if pneie" = an> then

I Pn C | ^ Pn+i ~ C >

so that the limit of pn, when n is infinite, is c.

Let cn = ceie«, a point on the singular circle, corresponding to an which is
assumed not to lie on it. Then, proceeding as in Weierstrass's theory in § 51,
if

«.= oo („ _

Gz= n

where gn(z) = - + L_ +... + _

Z-Cn 2 \Z-CnJ mn - I \ Z - Cn

G (z) is a uniform function, continuous everywhere in the plane except along
the circumference of the circle which may be a line of essential singularities.

Special simpler forms can be derived according to the character of the
series of quantities constituted by | an - cn . If there be a finite integer m,

00

such that 2 an — cn m is a converging series, then in gn (z) only the first

M = l

m — 1 terms need be retained.
Ex. Construct the function when

m being a given positive integer and r a positive quantity.

Again, the point cn was associated with an so that they have the same
argument : but this distribution of points on the circle is not necessary and
can be made in any manner which satisfies the condition that in the limited

00

case just quoted the series 2 an — cn m is a converging series.

Singular lines of other classes, for example, sectioiis\ in connection with functions
defined by integrals, arise in connection with analytical functions. They are discussed
by Painleve, "Sur les lignes singulieres des fonctions analytiques," (These, Gauthier-
Villars, Paris, 1887).

Ex. Shew that, if the zeros of a function be the points

. _b+c— (a — d) i

ZT ^ ~7 i 7T \ • 5

* This investigation is due to Picard, Comptes Rendus, t. xci, (1881), pp. 690—692.
t Called conpures by Hermite ; see § 103.

86.] LACUNARY FUNCTIONS 141

where a, ?;, c, d are integers satisfying the condition ad-bo = l, so that the function
has a circle of radius unity for an essential singular line, then if

b + di

„

2J = -^ - =— , ,

d+bi'

( A

the function n \ 5 e z

(z — li

where the product extends to all positive integers subject to the foregoing condition
ad-bc = l, is a uniform function finite for all points in the plane not lying on the
circle of radius unity. (Picard.)

87. In the earlier examples, instances were given of functions which
have only isolated points for their essential singularities : and, in the later
examples, instances have been given of functions which have lines of
essential singularities, that is, there are continuous lines for which the
functions do not exist. We now proceed to shew how functions can be
constructed which do not exist in assigned continuous spaces in the plane,
these spaces being aggregates of essential singularities. Weierstrass was
the first to draw attention to lacunary functions, as they may be called ;
the following investigation in illustration of Weierstrass's theorem is due to
Poincare' *.

Take any convex curve in the plane, say G ; and consider the function

*z^b'
where the quantities A are constants, subject to the conditions

(i) The series ^\A\ converges uniformly and unconditionally :
(ii) Each of the points b is either within or on the curve G :
(iii) The points b are the aggregate of all rational j points within and
on C : then the function is a uniform analytical function for all points
without C and it has the area of G for a lacunary space.

First, it is evident that, if z = b, then the series does not converge.
Moreover as the points b are the aggregate of all the rational points within
or on C, there will be an infinite number of singularities in the immediate
vicinity of b : we shall thus have an unlimited number of terms each infinite
of the first order, and thus (§ 42) the point b will be an essential singularity.
As this is true of all points z within or on C, it follows that the area C is a
lacunary space for the function, if the function exist at all.

Secondly, let z be a point without G ; and let d be the distance of z from
the nearest point of the boundary of C^f% so that d is not a vanishing quantity.

* Acta Soc. Fenn., t. xii, (1883), pp. 341—350.

J Rational points within or on C are points whose positions can be determined rationally in
terms of the coordinates of assigned points on C ; examples will be given.

t This will be either the shortest normal from z to the boundary or the distance of z from
some point of abrupt change of direction, as for instance at the angular point of a polygon.

142

FUNCTIONS WITH

[87.

Then | z — b \ ^ d ; and therefore

A _ \A\ \A\
~\z-b\< d '

z-b

so that

-b

A

z-b

Now 2 j.A| converges uniformly and unconditionally and therefore, as d does
not vanish,

z-b
converges uniformly and unconditionally, that is.

is a function of 2 which converges uniformly and unconditionally for every
point without C. Let it be denoted by <£ (z).

Let c be any point without C, and let r be the radius of the greatest
circle centre c which can be drawn so as to have no point of C within itself
or on its circumference, so that r is the radius of the domain of c; then
b — c > r, for all points b.

If we take a point z within this circle, we have \z — c =6r, where 6 < 1.

Now for all points within this circle the function <£ {z} converges uniformly,

A

and every term -- =• of <f> (z) is finite. Also, for points within the circle, we

A

can expand -- j in powers of z — c in the form

of a converging series. Hence, by § 82, we have

<£(*)= 2 Bm(z-c)m,

a series converging uniformly and unconditionally for all points within the
circle centre c and radius r, which circle is the circle of convergence of the
series. The function can be expressed in the usual manner over the whole of
the region of continuity, which is the part of the plane without the curve C.

Thus 0 (z) is a uniform analytical function, having the area of C for a
lacunary space.

As an example, take a convex polygon having o1} ...... , ap for its angular points;

then any point

...... +mj>ap

TOI + ...... +mp

where mlt ...... , mp are positive integers or zero (simultaneous zeros being excluded), is

87.] LACUNARY SPACES 143

either within the polygon or on its boundary : and any rational point within the polygon

or on its boundary can be represented by

p
2 mrar

r=l

P '
2 mr
r=l
by proper choice of ?n15 ...... , mp, a choice which can be made in an infinite number of ways.

Let ult ...... , Up be given quantities, the modulus of each of which is less than unity:

then the series

•9-11 m> 11 mf

«& ^ ' I ...... ftp

o

converges uniformly and unconditionally. Then all the assigned conditions are satisfied
for the function

_ .. . + mpap > '

ml + ...... +mp J

and therefore it is a function which converges uniformly and unconditionally everywhere
outside the polygon and which has the polygonal space (including the boundary) for
a lacunary space.

If, in particular, p = Z, we obtain a function which has the straight line
joining ax and a2 as a line of essential singularity. When we take at = 0,
a.2 = 1 and slightly modify the summation, we obtain the function

2 2 ^ 2

w=l m=0 W&

7i

which, when u^ <\ and |w2|<l, converges uniformly and unconditionally
everywhere in the plane except at points between 0 and 1 on the axis of real
quantities, this part of the axis being a line of essential singularity.
For the general case, the following remarks may be made :

(i) The quantities u1} u2>... need not be the same for every term; a
numerator, quite different in form, might be chosen, such as
(mj2+ ... + m/)"'1 where 2//, > p ; all that is requisite is that the
series, made up of the numerators, should converge uniformly
and unconditionally.

(ii) The preceding is only a particular illustration and is not necessarily
the most general form of function having the assigned lacunary
space.

It is evident that the first step in the construction of a function, which
shall have any assigned lacunary space, is the formation of some expression
which, by the variation of the constants it contains, can be made to
represent indefinitely nearly any point within or on the contour of the
space. Thus for the space between two concentric circles of radii a and c
and centre the origin we should take

Wja + O-WjU ^a«
-a£- e n

n

144 EXAMPLES [87.

which, by giving m^ all values from 0 to n, ra2 all values from 0 to n — 1 and
n all values from 1 to infinity will represent all rational points in the space :
and a function, having the space between the circles as lacunary, would be
given by

oo n n-1

2 2 2

n=l »»!=(> m2=0

(n — raj) b ^ 271-

•r /3

.6 — C

n

provided u\ < 1, u^ < 1, u2 < 1.

In particular, if a = 6, then the common circumference is a line of essential singularity
for the corresponding function. It is easy to see that the function

z — ae n

ao 2n-l m n

provided the series 2 2 u v

n=l m=0 m,n m, n

converges uniformly and unconditionally, is a function having the circle |0| = a as a line of
essential singularity.

Other examples will be found in memoirs by Goursat*, Poincaref, and HomenJ.

Ex. 1. Shew that the function

where r is a real positive quantity and the summation is for all integers m and n between
the positive and the negative infinities, is a uniform function in all parts of the plane
except the axis of real quantities which is a line of essential singularity.

Ex. 2. Discuss the region in which the function

w=i m=i jf/=i i ^- . ••- •
2—1 1 i

\7i 71

is definite. (Homen.)

Ex. 3. Prove that the function

n=0

exists only within a circle of radius unity and centre the origin. (Poincare.)

Ex. 4. An infinite number of points at, a2, as, are taken on the circumference of

a given circle, centre the origin, so that they form the aggregate of rational points on the
circumference. Shew that the series

2 l Z

can be expanded in a series of ascending powers of z which converges for points within the
circle, but that the function cannot be continued across the circumference of the circle.

(Stieltjes.)

* Comptes Rendus, t. xciv, (1882), pp. 715—718 ; Bulletin de Darboux, 2me Ser. , t. xi, (1887),
pp. 109—114.

t In the memoir, quoted p. 138, and Comptes Rendus, t. xcvi, (1883), pp. 1134-1136.
+ Acta Soc. Fenn., t. xii, (1883), pp. 445—464.

87.] EXAMPLES 145

Ex. 5. Prove that the series

2 | :

7T .00 -" K1-2TO-
9 oo

22

~li)2) '

7T _oo _oo (^(1 — 2wi — 2nz i) \zm-\-Anz~

where the summation extends over all positive and negative integral values of ra and of n
except simultaneous zeros, is a function which converges uniformly and unconditionally
for all points in the finite part of plane which do not lie on the axis of y ; and that
it has the value +1 or - 1 according as the real part of z is positive or negative.

(Weierstrass.)
Ex. 6. Prove that the region of continuity of the series

consists of two parts, separated by the circle z\ = l which is a line of infinities for
the series : and that, in these two parts of the plane, it represents two different
functions.

_<a'ir

If two complex quantities a> and to' be taken, such that z = e ^ and the real part of
^. is positive, and if they be associated with the elliptic function $ (u) as its half-periods,
then for values of z which lie within the circle z = \

in the usual notation of Weierstrass's theory of elliptic functions.

Find the function which the series represents for values of z without the circle \z\ = \.

(Weierstrass.)

Ex. 7. Four circles are drawn each of radius -^ having their centres at the points

1, i, - 1, -i respectively; the two parts of the plane, excluded by the four circumferences,
are denoted the interior and the exterior parts. Shew that the function

n='K sini^TT ( 1 1 1 1

is equal to IT in the interior part and is zero in the exterior part. (Appell.)

Ex. 8. Obtain the values of the function

»;-l- (-!)•(, i >1 l

«=i n V1 •> (2 + l)« (2-l)«J

in the two parts of the area within a circle centre the origin and radius 2 which lie

without two circles of radius unity, having their centres at the points 1 and - 1

respectively. (Appell.)

Ex- 9- If
and

,~3 ......

amr (2-«m)3 J

where the regions of continuity of the functions F extend over the whole plane, then / (z)
is a function existing everywhere except within the circles of radius unity described round
the points a, , «2, ...... , an. (Teixeira.)

F- 10

146 CLASSIFICATION [87.

Ex. 10. Let there be n circles having the origin for a common centre, and let
£,, (72, ...... , (7n, C'n + 1 be % + 1 arbitrary constants; also let a1} a2, ...... , an be any w points

lying respectively on the circumferences of the first, the second, ...... , the nth circles.

Shew that the expression

1 ("(CL
27T./0 W*

has the value <7m for points z lying between the (»w - l)th and the with circles and the
value (7n + 1 for points lying without the nth circle.

Construct a function which shall have any assigned values in the various bands into
which the plane is divided by the circles. (Pincherle.)

88. In § 32 it was remarked that the discrimination of the various
species of essential singularities could be effected by means of the properties
of the function in the immediate vicinity of the point.

Now it was proved, in § 63, that in the vicinity of an isolated essential
singularity b the function could be represented by an expression of the form

for all points in the space without a circle centre b of small radius and within
a concentric circle of radius not large enough to include singularities at
a finite distance from b. Because the essential singularity at b is isolated,
the radius of the inner circle can be diminished to be all but infinitesimal :

the series P (z — b) is then unimportant compared with G I —31 ) , which
can be regarded as characteristic for the singularity of the function.

Another method of obtaining a function, which is characteristic of the
singularity, is provided by § 68. It was there proved that, in the vicinity of
an essential singularity a, the function could be represented by an expression
of the form

where, within a circle of centre a and radius not sufficiently large to include
the nearest singularity at a finite distance from a, the function Q (z — a) is
finite and has no zeros : all the zeros of the given function within this circle
(except such as are absorbed into the essential singularity at a) are zeros of

the factor H ( - - ] , and the integer-index n is affected by the number of these
zeros. When the circle is made small, the function

z-a

can be regarded as characteristic of the immediate vicinity of a or, more
briefly, as characteristic of a.

88.] OF SINGULARITIES 147

It is easily seen that the two characteristic functions are distinct. For
if F and F^ be two functions, which have essential singularities at a of the
same kind as determined by the first characteristic, then

F(z)-Fl(z) = P(z-a)-Pl(z-a)
= P,(z-a\

while if their singularities at a be of the same kind as determined by the
second characteristic, then

F(z)_Q(z-a)
f\(*)-Q^-~a) = Q^2-

in the immediate vicinity of a, since Q1 has no zeros. Two such equations
cannot subsist simultaneously, except in one instance.

Without entering into detailed discussion, the results obtained in the
preceding chapters are sufficient to lead to an indication of the classification
of singularities*.

Singularities are said to be of the first class when they are accidental ;
and a function is said to be of the first class when all its singularities are of
the first class. It can, by § 48, have only a finite number of such singularities,
each singularity being isolated.

It is for this case alone that the two characteristic functions are in
accord.

When a function, otherwise of the first class, fails to satisfy the last
condition, solely owing to failure of finiteness of multiplicity at some point,
say at z = x , then that point ceases to be an accidental singularity. It has
been called (§ 32) an essential singularity ; it belongs to the simplest kind of
essential singularity ; and it is called a singularity of the second class.

A function is said to be of the second class when it has some singularities
of the second class ; it may possess singularities of the first class. By an
argument similar to that adopted in § 48, a function of the second class
can have only a limited number of singularities of the second class, each
singularity being isolated.

When a function, otherwise of the second class, fails to satisfy the last
condition solely owing to unlimited condensation at some point, say at z = oo ,
of singularities of the second class, that point ceases to be a singularity
of the second class: it is called a singularity (necessarily essential) of the
third class.

* For a detailed discussion, reference should be made to Guichard, " Theorie des points
singnhers essentiels" (These, Gauthier-ViUars, Paris, 1883), who gives adequate references to the

:stigations of Mittag-Leffler in the introduction of the classification and to the researches of
Cantor. See also Mittag-Leffler, Acta Math., t. iv, (1884), pp. 1_79; Cantor Crelle t Ixxxiv
1878), pp. 242—258, Acta Math., t. ii, (1883), pp. 311—328.

10—2

148 CLASSIFICATION OF SINGULARITIES [88.

A function is said to be of the third class when it has some singularities
of the third class ; it may possess singularities of the first and the second
classes. But it can have only a limited number of singularities of the third
class, each singularity being isolated.

Proceeding in this gradual sequence, we obtain an unlimited number of
classes of singularities: and functions of the various classes can be constructed
by means of the theorems which have been proved. A function of class n
has a limited number of singularities of class n, each singularity being
isolated, and any number of singularities of lower classes which, except in so
far as they are absorbed in the singularities of class n, are isolated points.

The effective limit of this sequence of classes is attained when the
number of the class increases beyond any integer, however large. When
once such a limit is attained, we have functions with essential singularities of
unlimited class, each singularity being isolated ; when we pass to functions
which have their essential singularities no longer isolated but, as in previous
class-developments, of infinite condensation, it is necessary to add to the
arrangement in classes an arrangement in a wider group, say, in species*.

Calling, then, all the preceding classes of functions functions of the first
species, we may, after Guichard (I.e.), construct, by the theorems already
proved, a function which has at the points al} a*,... singularities of classes
1, 2,..., both series being continued to infinity. Such a function is called
a function of the second species.

By a combination of classes in species, this arrangement can be continued
indefinitely ; each species will contain an infinitely increasing number of
classes; and when an unlimited number of species is ultimately obtained,
another wider group must be introduced.

This gradual construction, relative to essential singularities, can be carried
out without limit ; the singularities are the characteristics of the functions.

* Guichard (I.e.) uses the term genre.

CHAPTER VIII.

MULTIFORM FUNCTIONS.

89. HAVING now discussed some of the more important general properties
of uniform functions, we proceed to discuss some of the properties of multiform
functions.

Deviations from uniformity in character may arise through various causes :
the most common is the existence of those points in the ^-plane, which have
already (§ 12) been defined as branch-points.

As an example, consider the two power-series

Wl = l-i/-i/2-... , W2 = _(i_i/_^_... )f

which, for points in the plane such that z' is less than unity, are the two
values of (1 - /)* ; they may be regarded as two branches of the function w
defined by the equation

w2 = 1 — z' = z.

Let / describe a small curve (say a circle of radius r) round the point
z' = l, beginning on the axis of x\ the point 1 is the origin for z. Then z
is r initially, and at the end of the first description of the circle z is re2wi ;
hence initially wl is + 14 and w.2 is - r*} and at the end of the description
w1 is -f r^e™ and w2 is — r^e™, that is, wl is — rf and w.2 is + ri Thus the
effect of the single circuit is to change wl into w.2 and w2 into w1} that is,
the effect of a circuit round the point, at which w1 and w2 coincide in value,
is to interchange the values of the two branches.

If, however, z describe a circuit which does not include the branch-point,
wl and w2 return each to its initial value.

Instances have already occurred, e.g. integrals of uniform functions, in
which a variation in the path of the variable has made a difference in the

150 CONTINUATIONS [89.

result; but this interchange of value is distinct from any of the effects
produced by points belonging to the families of critical points which have
been considered. The critical point is of a new nature ; it is, in fact, a
characteristic of multiform functions at certain associated points.

We now proceed to indicate more generally the character of the relation
of such points to functions affected by them.

The method of constructing a monogenic analytic function, described in
§ 34, by forming all the continuations of a power-series, regarded as a given
initial element of the function, leads to the aggregate of the elements of the
function and determines its region of continuity. When the process of con
tinuation has been completely carried out, two distinct cases may occur.

In the first case, the function is such that any and every path, leading
from one point a to another point z by the construction of a series of
successive domains of points along the path, gives a single value at z as the
continuation of one initial value at a. When, therefore, there is only a
single value of the function at a, the process of continuation leads to only a
single value of the function at any other point in the plane. The function is
uniform throughout its region of continuity. The detailed properties of such
functions have been considered in the preceding chapters.

In the second case, the function is such that different paths, leading from
a to z, do not give a single value at z as the continuation of one and the
same initial value at a. There are different sets of elements of the function,
associated with different sets of consecutive domains of points on paths from
a to z, which lead to different values of the function at z; but any change
in a path from a to z does not necessarily cause a change in the value of the
function at z. The function is multiform in its region of continuity. The
detailed properties of such functions will now be considered.

90. In order that the process of continuation may be completely carried
out, continuations must be effected, beginning at the domain of any point a
and proceeding to the domain of any other point b by all possible paths in
the region of continuity, and they must be effected for all points a and b.
Continuations must be effected, beginning in the domain of every point a
and returning to that domain by all possible closed paths in the region of
continuity. When they are effected from the domain of one point a to that
of another point b, all the values at any point z in the domain of a (and not
merely a single value at such points) must be continued : and similarly when
they are effected, beginning in the domain of a and returning to that domain.
The complete region of the plane will then be obtained in which the function
can be represented by a series of positive integral powers : and the boundary
of that region will be indicated.

90.] OF A MULTIFORM FUNCTION 151

In the first instance, let the boundary of the region be constituted by a
number, either finite or infinite, of
isolated points, say L1} L2, Ls, ...
Take any point A in the region, so
that its distance from any of the
points L is not infinitesimal ; and
in the region draw a closed path
ABC...EFA so as to enclose one
point, say Ll} but only one point, of
the boundary and to have no point

of the curve at a merely infinitesimal distance from L^ Let such curves be
drawn, beginning and ending at A, so that each of them encloses one and
only one of the points of the boundary : and let Kr be the curve which
encloses the point Lr.

Let Wj be one of the power-series defining the function in a domain with
its centre at A : let this series be continued along each of the curves Ks by
successive domains of points along the curve returning to A. The result
of the description of all the curves will be that the series w^ cannot be
reproduced at A for all the curves though it may be reproduced for some
of them ; otherwise, w: would be a uniform function. Suppose that w.2, w3> ...,
each in the form of a power-series, are the aggregate of new distinct values
thus obtained at A ; let the same process be effected on w2, w3, ... as has
been effected on w1; and let it further be effected on any new distinct values
obtained at A through w2, w3, ... , and so on. When the process has
been carried out so far that all values obtained at A, by continuing any
series round any of the curves K back to A, are included in values already
obtained, the aggregate of the values of the function at A is complete : they
are the values at A of the branches of the function.

We shall now assume that the number of values thus obtained is finite,
say n, so that the function has n branches at A : if their values be denoted
by w1} w2, ..., wn, these n quantities are all the values of the function at A.
Moreover, n is the same for all points in the plane, as may be seen by con
tinuing the series at A to any other point and taking account of the corollaries
at the end of the present section.

The boundary-points L may be of two kinds. It may (and not infre
quently does) happen that a point Ls is such that, whatever branch is taken
at A as the initial value for the description of the circuit Ks, that branch is
reproduced at the end of the circuit. Let the aggregate of such points be
/u J2, .... Then each of the remaining points L is such that a description
of the circuit round it effects a change on at least one of the branches, taken
as an initial value for the description ; let the aggregate of these points be
Blt 52, .... They are the branch-points; their association with the definition
in § 12 will be made later.

152

DEFORMATION OF PATH

[90.

Fig. 15.

When account is taken of the continuations of the function from a point
A to another point B, we have n values at B as the continuations of n values
at A. The selection of the individual branch at B, which is the continuation
of a particular branch at A, depends upon the path of z between A and 5;
it is governed by the following fundamental proposition : —

The final value of a branch of a function for two paths of variation of the
independent variable from one point to another will be the same, if one path
can be deformed into the other without passing over a branch-point.

Let the initial and the final points be a and b, and let one path of
variation be acb. Let another path of variation be aeb,
both paths lying in the region in which the function can
be expressed by series of positive integral powers : the two
paths are assumed to have no point within an infinitesimal
distance of any of the boundary-points L and to be taken
so close together that the circles of convergence of pairs of
points (such as cx and e1} c2 and e2, and so on) along the two
paths have common areas. When we begin at a with a
branch of the function, values at d and at e^ are obtained,
depending upon the values of the branch and its derivatives at a and upon
the positions of ca and e^ hence, at any point in the area common to the
circles of convergence of these two points, only a single value arises as
derived through the initial value at a. Proceeding in this way, only a single
value is obtained at any point in an area common to the circles of con
vergence of points in the two paths. Hence ultimately one and the same
value will be obtained at b as the continuation of the value of the one branch
at a by the two different paths of variation which have been taken so that
no boundary-point L lies between them or infinitesimally near to them.

Now consider any two paths from a to b, say acb and adb, such that
neither of them is near a boundary-point and that the
contour they constitute does not enclose a boundary-point.
Then by a series of successive infinitesimal deformations we
can change the path acb to adb ; and as at b the same value
of w is obtained for variations of z from a to b along the
successive deformations, it follows that the same value of w
is obtained at b for variations of z along acb as for varia
tions along adb.

Next, let there be two paths acb, adb constituting a closed contour,
enclosing one (but not more than one) of the points / and none of the points
B. When the original curve K which contains the point / is described, the
initial value is restored : and hence the branches of the function obtained at
any point of K by the two paths from any point, taken as initial point, are
the same. By what precedes, the parts of this curve K can be deformed

Fig. 16.

90.] OF THE VARIABLE 153

into the parts of acbda without affecting the branches of the function : hence
the value obtained at b, by continuation along acb, is the same as the value
there obtained by continuation along adb. It therefore follows that a path
between two points a and b can be deformed over any point / without
affecting the value of the function at b ; so that, when the preceding
results are combined, the proposition enunciated is proved.

By the continued application of the theorem, we are led to the following
results : —

COROLLARY I. Whatever be the effect of the description of a circuit on the
initial value of a function, a reversal of the circuit restores the original value
of the function.

For the circuit, when described positively and negatively, may be re
garded as the contour of an area of infinitesimal breadth, which encloses no
branch-point within itself and the description of the contour of which
therefore restores the initial value of the function.

COROLLARY II. A circuit can be deformed into any other circuit without
affecting the final value of the function, provided that no branch-point be crossed
in tJie process of deformation.

It is thus justifiable, and it is often convenient, to deform a path con
taining a single branch-point into a loop round the
point. A loop* consists of a line nearly to the point, °~
nearly the whole of a very small circle round the point, Fig. 17.

and a line back to the initial point; see figure 17.

COROLLARY III. The value of a function is unchanged when the variable
describes a closed circuit containing no branch-point ; it is likewise unchanged
when the variable describes a closed circuit containing all the branch-points.

The first part is at once proved by remarking that, without altering the
value of the function, the circuit can be deformed into a point.

For the second part, the simplest plan is to represent the variable on
Neumann's sphere. The circuit is then a curve on the sphere enclosing all
the branch-points : the effect on the value of the function is unaltered by any
deformation of this curve which does make it cross a branch-point. The
curve can, without crossing a branch-point, be deformed into a point in that
other part of the area of the sphere which contains none of the branch
points ; and the point, which is the limit of the curve, is not a branch
point. At such a point, the value of the function is unaltered ; and there
fore the description of a circuit, which encloses all the branch-points,
restores the initial value of the function.

COROLLARY IV. If the values of w at b for variations along two paths

* French writers use the word lacet, German writers the word Schleife.

154

EFFECT OF DEFORMATION

[90.

acb, adb be not the same, then a description of acbda will not restore the initial
value of w at a.

In particular, let the path be the loop OeceO (fig. 17), and let it change w
at 0 into w'. Since the values of w at 0 are different and because there is
no branch-point in Oe (or in the evanescent circuit OeO), the values of w at
e cannot be the same : that is, the value with which the infinitesimal circle
round a begins to be described is changed by the description of that circle.
Hence the part of the loop that is effective for the change in the value of w is
the small circle round the point ; and it is because the description of a small
circle changes the value of w that the value of w is changed at 0 after the
description of a loop.

If/0?) be the value of w which is changed mtof^z) by the description of
the loop, so that/Oz) and f^(z) are the values at 0, then the foregoing
explanation shews that /(e) and / (e) are the values at e, the branch /(e)
being changed by the description of the circle into the branch /i(e).

From this result the inference can be derived that the points Bl} B.2, ...
are branch-points as defined in § 12. Let a be any one of the points, and
let f(z) be the value of w which is changed into f, (z) by the description of
a very small circle round a. Then as the branch of w is monogenic, the
difference between f(z) and f^(z) is an infinitesimal quantity of the same
order as the length of the circumference of the circle : so that, as the circle
is infinitesimal and ultimately evanescent, \f(z) -/iOz)| can be made as small
as we please with decrease of z - a or, in the limit, the values of /(a) and
/(a) at the branch-point are equal. Hence each of the points B is such
that two or more branches of the function have the same value at the point
and there is interchange among these branches when the variable describes a
small circuit round the point : which affords a definition of a branch-point,
more complete than that given in § 12.

COROLLARY V. If a closed circuit contain several branch-points, the effect
which it produces can be obtained by a combination of the effects produced in
succession by a set of loops each going round only one of the branch-points.

If the circuit contain several branch-points, say three as at a, b, c, then a
path such as AEFD, in fig. 18, can without
crossing any branch-point, be deformed into the
loops AaB, BbC, GcD; and therefore the complete
circuit AEFD A can be deformed validly into
AaBbCcDA, and the same effect will be produced
by the two forms of circuit. When D is made DA

practically to coincide with A, the whole of the Fig. 18.

second circuit is composed of the three loops. Hence the corollary.

This corollary is of especial importance in the consideration of integrals
of multiform functions.

91.] OF PATH OF THE VAKIABLE 155

COROLLARY VI. In a continuous part of the plane where there are no
branch-points, each branch of a multiform function is uniform.

Each branch is monogenic and, except at isolated points, continuous;
hence, in such regions of the plane, all the propositions which have been
proved for monogenic analytic functions can be applied to each of the
branches of a multiform function.

91. If there be a branch-point within the circuit, then the value of the
function at 6 consequent on variations along acb may, but will not necessarily,
differ from its value at the same point consequent on variations along adb.
Should the values be different, then the description of the whole curve acbda
will lead at a not to the initial value of w, but to a different value.
The test as to whether such a change is effected by the description is
immediately derivable from the foregoing proposition; and as in Corollary
IV., § 90, it is proved that the value is or is not changed by the loop,
according as the value of w for a point near the circle of the loop is
or is not changed by the description of that circle. Hence it follows that, if
there be a branch-point which affects the branch of the function, a path of
variation of the independent variable cannot be deformed across the branch
point without a change in the value of w at the extremity of the path.

And it is evident that a point can be regarded as a branch-point for a
function only if a circuit round the point interchange some (or all) of the
branches of the function which are equal at the point. It is not necessary that
all the branches of the function should be thus affected by the point : it is
sufficient that some should be interchanged*.

Further, the change in the value of w for a single description of a circuit
enclosing a branch-point is unique.

For, if a circuit could change w into w' or w", then, beginning with w"
and describing it in the negative sense we should return to w and afterwards
describing it in the positive sense with w as the initial value we should
obtain w'. Hence the circuit, described and then reversed, does not restore
the original value w" but gives a different branch w' ; and no point on
the circuit is a branch-point. This result is in opposition to Corollary I.,
of § 90 ; and therefore the hypothesis of alternative values at the end of
the circuit is not valid, that is, the change for a single description is
unique.

But repetitions of the circuit may, of course, give different values at the
end of successive descriptions.

* In what precedes, certain points were considered which were regular singularities (see
p. 163, note) and certain which were branch-points. Frequently points will occur which are at
once branch-points and infinities ; proper account must of course be taken of them.

156

LAW OF INTERCHANGE

[92.

Fig. 19.

92. Let 0 be any ordinary point of the function ; join it to all the
branch-points (generally assumed finite in
number) in succession by lines which do not
meet each other : then each branch is uniform
for each path of variation of the variable which
meets none of these lines. The effects pro
duced by the various branch-points and their
relations on the various branches can be indi
cated by describing curves, each of which
begins at a point indefinitely near 0 and
returns to another point indefinitely near it
after passing round one of the branch- points,
and by noting the value of each branch of the function after each of these
curves has been described.

The law of interchange of branches of a function after description of a
circuit round a branch-point is as follows: —

All the branches of a function, which are affected by a branch-point as such,
can either be arranged so that the order of interchange (for description of a
path round the point) is cyclical, or be divided into sets in each of which the
order of interchange is cyclical.

Let wlt w.2> w3)... be the branches of a function for values of z near a
branch-point a which are affected by the description of a small closed curve
C round a : they are not necessarily all the branches of the function, but only
those affected by the branch-point.

The branch w^ is changed after a description of C ; let w2 be the branch
into which it is changed. Then w2 cannot be unchanged by C; for a reversed
description of C, which ought to restore w1} would otherwise leave w.2 un
changed. Hence w2 is changed after a description of (7; it may be changed
either into w1 or into a new branch, say w3. If into wlt then w-^ and w2 form
a cyclical set.

If the change be into w3, then w3 cannot remain unchanged after a
description of C, for reasons similar to those that before applied to the
change of w.2: and it cannot be changed into w2, for then a reversed de
scription of G would change wz into w.A, and it ought to change w2 into w^
Hence, after a description of C, w3 is changed either into w^ or into a new
branch, say w4. If into w1} then w1} w2, w3 form a cyclical set.

If the change be into w4, then w4 cannot remain unchanged after a
description of G ; and it cannot be changed into w.2 or ws, for by a reversal
of the circuit that earlier branch would be changed into w4 whereas it ought
to be changed into the branch, which gave rise to it by the forward descrip
tion — a branch which is not w4. Hence, after a description of C, w4 is
changed either into w^ or into a new branch. If into wlf then wj} w.2, w3, w4
form a cyclical set.

92.] OF BRANCHES OF A FUNCTION 157

If w4 be changed into a new branch, we proceed as before with that new
branch and either complete a cyclical set or add one more to the set. By
repetition of the process, we complete a cyclical set sooner or later.

If all the branches be included, then evidently their complete system
taken in the order in which they come in the foregoing investigation is a
system in which the interchange is cyclical.

If only some of the branches be included, the remark applies to the set
constituted by them. We then begin with one of the branches not included
in that set and evidently not inclusible in it, and proceed as at first, until
we complete another set which may include all the remaining branches or
only some of them. In the latter case, we begin again with a new branch
and repeat the process ; and so on, until ultimately all the branches are
included. The whole system is then arranged in sets, in each of which the
order of interchange is cyclical.

93. The analytical test of a branch-point is easily obtained by con
structing the general expression for the branches of a function which are
interchanged there.

Let z = a be a branch-point where n branches w1} ^v2,..., wn are cyclically
interchanged. Since by a first description of a small curve round a, the
branch w1 changes into w2, the branch w» into ws, and so on, it follows that
by r descriptions w1 is changed into wr+l and by n descriptions wl reverts to
its initial value. Similarly for each of the branches. Hence each branch
returns to its initial value after n descriptions of a circuit round a branch
point where n branches of the function are interchangeable.

Now let z - a = Zn ;

then, when z describes circles round a, Z moves in a circular arc round its
origin. For each circumference described by z, the variable Z describes

-th part of its circumference; and the complete circle is described by Z
round its origin when n complete circles are described by z round a. Now
the substitution changes wr as a function of z into a function of Z, say into
Wr; and, after n complete descriptions of the ^-circle round a, wr returns
to its initial value. Hence, after the description of a ^-circle round its
origin, Wr returns to its initial value, that is, Z = 0 ceases to be a branch
point for Wr. Similarly for all the branches W.

But no other condition has been associated with a as a point for the
function w ; and therefore Z = 0 may be any point for the function W, that
is, it may be an ordinary point, or a singularity. In every case we have W
a uniform function of Z in the immediate vicinity of the origin ; and therefore
in that vicinity it can be expressed in the form

158 ANALYTICAL EXPRESSION [93.

with the significations of P and G already adopted. When Z is an ordinary
point, G is a constant or zero ; when Z is an accidental singularity, O is an
algebraical function ; and, when Z is an essential singularity, G is a transcen
dental function.

The simpler cases are, of course, those in which the form of G is alge
braical or constant or zero ; and then W can be put into the form

ZmP(Z),

where P is an infinite series of positive powers and m is an integer. As this
is the form of W in the vicinity of Z=Q, it follows that the form of w in the
vicinity of z = a is

m 1

(z - a)n P {(z - a)n}

and the various n branches of the function are easily seen to be given by

i
substituting in the above for (z — a)n the values

2im j.

e m (z — of,

where s = 0, 1,..., n — 1. We therefore infer that the general expression for
the n branches of a function, which are interchanged by circuits round a
branch-point z = a, assumed not to be an essential singularity, is

m _ 1

(z - a)Tl P {(z - a)»},

i

where m is an integer, and where to (z — a)n its n values are in turn assigned
to obtain the different branches of the function.

There may be, however, more than one cyclical set of branches. If there
be another set of r branches, then it may similarly be proved that their

general expression is

OTI _ i

(zjaYQ{(z-ay-},

where m^ is an integer, and Q is an integral function ; the various branches

i
are obtained by assigning to (z — a)r its r values in turn.

And so on, for each of the sets, the members of which are cyclically
interchangeable at the branch-point.

When the branch-point is at infinity, a different form is obtained. Thus
in the case of a set of n cyclically interchangeable branches we take

z = %-»,

so that n negative descriptions of a closed £-curve, excluding infinity and no
other branch-point, requires a single positive description of a closed curve
round the w-origin. These n descriptions restore the value of w; as a function
of z to its initial value; and therefore the single description of the M- curve
round the origin restores the value of U — the equivalent of w after the

93.] NEAR A BRANCH-POINT 159

change of the independent variable — as a function of u. Thus u = 0 ceases
to be a branch-point for the function U ; and therefore the form of U is

. .

where the symbols have the same general signification as before.

If, in particular, z = oo be a branch-point but not an essential singularity,
then G is either a constant or an algebraical function ; and then U can be
expressed in the form

u~mP(u},

where TO is an integer. When the variable is changed from u to z, then the
general expression for the n branches of a function which are interchangeable

at z = oo , assumed not to be an essential singularity, is

where TO is an integer and where to zn its n values are assigned to obtain the
different branches of the function.

If, however, the branch-point z = a in the former case or z = oo in the
latter be an essential singularity, the forms of the expressions in the vicinity
of the point are

_i i

G{(z-a) »J>-p{(jr-aJ»}f

i _i

and G(zn) + P(z n),

respectively.

Note. When a multiform function is denned, either explicitly or im
plicitly, it is practically always necessary to consider the relations of the
branches of the function for z = oo as well as their relations for points that
are infinities of the function. The former can be determined by either
of the processes suggested in § 4 for dealing with z=<x>; the latter can be
determined as in the present article.

Moreover, the total number of branches of the function has been assumed
to be finite. The cases, in which the number of branches is unlimited, need
not be discussed in general : it will be sufficient to consider them when they
arise, as they do arise, e.g., when the function is of the form of an algebraical
irrational with an irrational index such as z^ — hardly a function in the
ordinary sense—, or when the function is the logarithm of a function of z,
or is the inverse of a periodic function. In the nature of their multiplicity
of branching and of their sequence of interchange, they are for the most part
distinct from the multiform functions with only a finite number of branches.

Ex. The simplest illustrations of multiform functions are furnished by functions
denned by algebraical equations, in particular, by algebraic irrationals.

160

ALGEBRAICAL

[93.

The general type of the algebraical irrational is the product of a number of functions
of the form w = {A(z — al)(z-a.2) ...... (z-a^)}m, m and n being integers.

This particular function has m branches; the points a1} «2, ...... , an are branch-points.

To find the law of interchange, we take z-ar = pe01; then when a small circle of radius p
is described round ar, so that z returns to its initial position, the value of 6 increases by

2n and the new value of w is aw, where a is the with root of unity defined by em m. Taking
then the various branches as given by w, aw, a?w, ...... , am~lw, we have the law of inter

change for description of a small curve round any one-branch point as given by this
succession in cyclical order. The law of succession for a circuit enclosing more than
one of the branch-points is derivable by means of Corollary V, § 90.

To find the relation of z = o> to w, we take zz' = l and consider the new function W in
the vicinity of the ^'-origin. We have

W ={A (1 -VH1 -a/) ...... (l-an<)}^'~»».

If the variable z1 describe a very small circle round the origin in the negative sense, then

27TZ —

z' is multiplied by e~2™ and so W acquires a factor e ™, that is, W is changed unless
this acquired factor is unity. It can be unity only when n/m is an integer ; and therefore,
except when n/m is an integer, 0=00 is a branch-point of the function. The law of
succession is the same as that for negative description of the z'-circle, viz., w, anw,
a2nw, ...... ; the m values form a single cycle only if n be prime to m, and a set of cycles

if n be not prime to m.

Thus 0=00 is a branch -point for w = (k?-gg-g^~^ ; it is not a branch-point for
w = {(\ -22) (1 — &2z2)}~*; and z = b is a branch-point for the function defined by

(z — b) w2 = z — a,
but z = b is riot a branch-point for the function defined by (z—b)2wz = z-a.

Again, if p denote a particular value of ft when z has a given value, and q similarly

denote a particular value of [— — : ) , then w=p+q is a six-valued function, the values

V+v
being

W6= -p + aq,

where a is a primitive cube root of unity. The branch-points are - 1, 0, 1, oo ; and the
orders of change for small circuits round one (and only one) of these points are as
follows :

For a small circuit round

-1

0

1

00

Wj changes to

•ft

W-i

W3

W2

W2 „

we

W1

W4

w,

^3

to,

W4

W5

W4

W4 „

M>2

w,

W6

W3

Ws n

W3

W6

w,

W6

U>6

W74

W5

Wo

W5

93.] FUNCTIONS 161

Combinations can at once be effected ; thus, for a positive circuit enclosing both 1 and QO
but* not — 1 or 0, the succession is

iolt w4, w6, w2, w3, WG
in cyclical order.

94. It has already been remarked that algebraic irrationals are a special
class of functions denned by algebraical equations. Functions thus generally
denned by equations, which are algebraical so far as concerns the dependent
variable but need not be so in reference to the independent variable, are
often called algebraical. The term, in one sense, cannot be strictly applied
to the roots of an equation of every degree, seeing that the solution
of equations of the fifth and higher degrees can be effected only by
transcendental functions; but what is implied is that a finite number of
determinations of the dependent variable is given by the equation -f*.

The equation is algebraical in relation to the dependent variable w, that
is, it will be taken to be of finite degree n in w. The coefficients of the
different powers will be supposed to be rational uniform functions of z : were
they irrational in any given equation, the equation could be transformed
into another, the coefficients of which are rational uniform functions. And
the equation is supposed to be irreducible, that is, if the equation be taken
in the form

f(w, *) = 0,

the left-hand member f(w, z) cannot be resolved into factors of a form and
character as regards w and z similar to /itself.

The existence of equal roots of the equation for general values of z
requires that

fi \ j "df(w> z)
f(w,z) and ^~

shall have a common factor, which will be rational owing to the form of
f(w, z}. This form of factor is excluded by the irreducibility of the equation ;
so that /= 0, as an equation in w, has not equal roots for general values
of z. But though the two equations are not both satisfied in virtue of a
simpler equation, they are two equations determining values of w and #;
and their form is such that they will give equal values of w for special
values of z.

Since the equation is of degree n, it may be taken to be

w

where the functions F1} F2}... are rational and uniform. If all their singu-

* Such a circuit, if drawn on the Neumann's sphere, may be regarded as excluding - 1 and 0,
or taking account of the other portion of the surface of the sphere, it may be regarded as a
negative circuit including - 1 and 0, the cyclical interchange for which is easily proved to be
iCj, w4, w5, w.2, M?3, w6 as in the text.

t Such a function is called Men defini by Liouville.

F. 11

162 ALGEBRAICAL [94.

larities be accidental, they are raeromorphic algebraical functions of z (unless
z = oo is the only singularity, in which case they are holomorphic) ; and the
equation can then be replaced by one which is equivalent and has all its
coefficients holomorphic, the coefficient of wn being the least common multiple
of all the denominators of the meromorphic functions in the first form. This
form cannot however be deduced, if any of the singularities be essential.

The equation, as an equation in w, has n roots, all functions of z ; let
these be denoted by w1,w2,..., ivn, which are the n branches of the function w.
When the geometrical interpretation is associated with the analytical relation,
there are n points in the w-plane, say a1,..., an, which correspond with a point
in the ^-plane, say with c^ ; and in general these n points are distinct. As
z varies so as to move in its own plane from a, then each of the w-points
moves in their common plane ; and thus there are n w-paths corresponding
to a given z-path. These n curves may or may not meet one another.

If they do not, there are n distinct w-paths, leading from a1;..., an to
/3i,..., /3n, respectively corresponding to the single ^-path leading from a
to b.

If two or more of the w-paths do meet one another, and if the describing
w-poirits coincide at their point of intersection, then at such a point of
intersection in the w-plane, the associated branches w are equal ; and
therefore the point in the ^-plane is a point that gives equal values for w.
It is one of the roots of the equation obtained by the elimination of w
between

the analytical test as to whether the point is a branch-point will be
considered later. The march of the concurrent ^-branches from such a
point of intersection of two w-paths depends upon their relations in its
immediate vicinity.

When no such point lies on a ^-path from a to b, no two of the w-points
coincide during the description of their paths. By § 90, the 2-path can be
deformed (provided that, in the deformation, it does not cross a branch-point)
without causing any two of the w-points to coincide. Further, if z describe
a closed curve which includes none of the branch -points, then each of the
^-branches describes a closed curve and no two of the tracing points ever
coincide.

Note. The limitation for a branch-point, that the tracing w-points
coincide at the point of intersection of the w-curves, is of essential im
portance.

What is required to establish a point in the z-plane as a branch-point,
is not a mere geometrical intersection of a couple of completed w-paths but
the coincidence of the w-points as those paths are traced, together with inter-

94.] FUNCTIONS 1 63

change of the branches for a small circuit round the point. Thus let there be
such a geometrical intersection of two w-curves, without coincidence of the
tracing points. There are two points in the ^-plane corresponding to the
geometrical intersection ; one belongs to the intersection as a point of the
w-paih which first passed through it, and the other to the intersection as a
point of the w-path which was the second to pass through it. The two
branches of w for the respective values of z are undoubtedly equal ; but the
equality would not be for the same value of z. And unless the equality
of branches subsists for the same value of z, the point is not a branch
point.

A simple example will serve to illustrate these remarks. Let w be defined by the
equation

so that the branches w1 and w2 are given by

Ci0j_ = cz+z(z2 + c2)*, cw2 = cz-z(z*-\- c2)* ;
it is easy to prove that the equation resulting from the elimination of w between /=0 and

and that only the two points z= ±ic are branch-points.

The values of z which make wl equal to the value of wz for z = a (supposed not equal to
either 0, ci or — ci) are given by

cz + z (02 + c2)* = ca - a (a2 + c2)*,

which evidently has not 2 = a for a root. Rationalising the equation so far as concerns z
and removing the factor z -a, as it has just been seen not to furnish a root, we find that s
is determined by

z3 + z2a + za2 + a3 + 2ac2 - 2ac (a2 + c2) * = 0,

the three roots of which are distinct from a, the assumed point, and from ±ci, the branch
point. Each of these three values of z will make wv equal to the value of w2 for z=a : we
have geometrical intersection without coincidence of the tracing points.

95. When the characteristics of a function are required, the most im
portant class are its infinities: these must therefore now be investigated.
It is preferable to obtain the infinities of the function rather than the
singularities alone, in the vicinity of which each branch of the function
is uniform * : for the former will include these singularities as well as
those branch-points which, giving infinite values, lead to regular singularities
when the variables are transformed as in § 93. The theorem which deter
mines them is: —

The infinities of a function determined by an algebraical equation are the
singularities of the coefficients of the equation.
Let the equation be

wn + wn-i FI ^ + wn-, !»,(*) + ... + rf^ (^) + ^ (^) = Q,

* These singularities will, for the sake of brevity, be called regular.

11—2

164 INFINITIES [95.

and let w' be any branch of the function; then, if the equation which
determines the remaining branches be

wn-i + wn-2 Qi ^ + wn-3 £2 (Y) + . . . + WGn-z (Z) + Gn-i (z) = 0,

we have Fn (z) = - w'Gn-i (z),

Fn^ (z) = - w'Gn-z (z) + £„_! (z),

^71-2 (Z) = - w'Gn-s (z} + #n-2 (z),

Now suppose that a is an infinity of w' ; then, unless it be a zero of order
at least equal to that of Gn^ (z), a is an infinity of Fn (z). If, however, it be
a zero of Gn-i (z) of sufficient order, then from the second equation it is an
infinity of Fn_l(z) unless it is a zero of order at least equal to that of
6rn_2 (z) ; and so on. The infinity must be an infinity of some coefficient not
earlier than Fi (z) in the equation, or it must be a zero of all the functions
G which are later than Gf_! (z). If it be a zero of all the functions Gr, so
that we may not, without knowing the order, assert that it is of rank at
least equal to its order as an infinity of w', still from the last equation it
follows that a must be an infinity of Fl (z). Hence any infinity of w is an
infinity of at least one of the coefficients of the equation.

Conversely, from the same equations it follows that a singularity of one
of the coefficients is an infinity either of w' or of at least one of the co
efficients G. Similarly the last alternative leads to an inference that the
infinity is either an infinity of another branch w" or of the coefficients of the
(theoretical) equation which survives when the two branches have been
removed. Proceeding in this way, we ultimately find that the infinity either
is an infinity of one of the branches or is an infinity of the coefficient in the
last equation, that is, of the last of the branches. Hence any singularity
of a coefficient is an infinity of at least one of the branches of the function.

It thus appears that all the infinities of the function are included among,
and include, all the singularities of the coefficients ; but the order of the
infinity for a branch does not necessarily make that point a regular
singularity nor, if it be made a regular singularity, is the order necessarily
the same as for the coefficient.

96. The following method is effective for the determination of the order
of the infinity of the branch.

Let a be an accidental singularity of one or more of the F functions,
say of order ra; for the function Ft ; and assume that, in the vicinity of a,
we have

Ft (z) = (z- a)-™* [Ci + di (z-a) + e{ (z - a? +...].

96.]

OF ALGEBRAICAL FUNCTIONS

165

Then the equation which determines the first term of the expansion of w in
a series in the vicinity of a is

wn + d (z — a)~™i wn~l + c2(z — a)~m2 wn~2 + ...

-f cn_! (z - a)~m»-i w + cn (z - a)~m« = 0.

Mark in a plane, referred to two rectangular axes, points n, 0; n — 1,
— m^; n — 2, — m2 ; . . ., 0, — mn ; let these
be A0, A1} ..., An respectively. Any line
through Ai has its equation of the form

1 1 —I— nm • ~~ ~\ J o" — l w — T. t)\\
y T »*< — A, {J, (71, tftf

that is,

y — \x = — \ (n — i) — mi.

If then w = (z — a)~xf(z}, where f(z) is
finite when z = a, the intercept of the fore
going line on the negative side of the axis of y is equal to the order of the
infinity in the term

wn-iFi(z).

This being so, we take a line through An coinciding in direction with the
negative part of the axis of y and we turn it about An in a trigonometrically
positive direction until it first meets one of the other points, say An_r ; then
we turn it about An_r until it meets one of the other points, say An_s; and
so on until it passes through A0. There will thus be a line from An to
A0, generally consisting of a number of parts ; and none of the points A
will be outside it.

The perpendicular from the origin on the line through An_r and An_g is
evidently greater than the perpendicular on any parallel line through a
point A, that is, on any line through a point A with the same value
of X; and, as this perpendicular is

it follows the order of the infinite terms in the equation, when the particular
substitution is made for w, is greater for terms corresponding to points lying
on the line than it is for any other terms.

If /(*) = 0 wnen z = a, then the terms of lowest order after the substitu
tion of (z — a)~Kf(z) for w are

as many terms occurring in the bracket as there are points A on the line
joining An_r to An_s. Since the equation determining w must be satisfied,
terms of all orders must disappear, and therefore

an equation determining s-r values of 6, that is, the first terms in the
expansions of s — r branches w.

166 INFINITIES [96.

Similarly for each part of the line : for the first part, there are r branches
with an associated value of X ; for the second, s — r branches with another
associated value ; for the third, t — s branches with a third associated value ;
and so on.

The order of the infinity for the branches is measured by the tangent
of the angle which the corresponding part of the broken line makes with the
axis of a; ; thus for the line joining An^. to An_s the order of the infinity for
the s - r branches is

where mn_r and mn_s are the orders of the accidental singularities of Fn_r (z)
and Fn_s (z).

If any part of the broken line should have its inclination to the axis of
x greater than \ir so that the tangent is negative and equal to - //,, then the
form of the corresponding set of branches w is (z — a,y g {z} for all of them,
that is, the point is not an infinity for those branches. But when the
inclination of a part of the line to the axis is < \TT, so that the tangent is
positive and equal to X, then the form of the corresponding set of branches
w is (z — a)~Kf(z) for all of them, that is, the point is an infinity of order X
for those branches.

In passing from An to A0 there may be parts of the broken line which
have the tangential coordinate negative, implying therefore that a is not an
infinity of the corresponding set or sets of branches w. But as the revolving
line has to change its direction from Any' to some direction through A0,
there must evidently be some part or parts of the broken line which have
their tangential coordinate positive, implying therefore that a is an infinity
of the corresponding set or sets of branches.

Moreover, the point a is, by hypothesis, an accidental singularity of at
least one of the coefficients and it has been supposed to be an essential
singularity of none of them; hence the points A0, A1} ..., An are all in the
finite part of the plane. And as no two of their abscissa are equal, no line
joining two of them can be parallel to the axis of y, that is, the inclination
of the broken line is never \ir and therefore the tangential coordinate is
finite, that is, the order of the infinity for the branches is finite for any
accidental singularity of the coefficients.

If the singularity at a be essential for some of the coefficients, the
corresponding result can be inferred by passing to the limit which is
obtained by making the corresponding value or values of m infinite. In
that case the corresponding points A move to infinity and then parts of the
broken line pass through A0 (which is always on the axis of x) parallel to
the axis of y, that is, the tangential coordinate is infinite and the order of

96.] OF ALGEBRAICAL FUNCTIONS 167

the infinity at a for the corresponding branches is also infinite. The point is
then an essential singularity (and it may be also a branch-point).

It has been assumed implicitly that the singularity is at a finite point in
the 2-plane ; if, however, it be at oo , we can, by using the transformation
zz' — 1 and discussing as above the function in vicinity of the origin, obtain
the relation of the singularity to the various branches. We thus have the
further proposition :

The order of ike infinity of a branch of an algebraical function at a
singularity of a coefficient of the equation, which determines the function, is
finite or infinite according as the singularity is accidental or essential.

If the coefficients FI of the equation be holomorphic functions, then
z = oo is their only singularity and it is consequently the only infinity for
branches of the function. If some of or all the coefficients Ff be mero-
morphic functions, the singularities of the coefficients are the zeros of
the denominators and, possibly, £=oo; and, if the functions be algebraical,
all such singularities are accidental. In that case, the equation can be
modified to

h0 (z) wn + h^ (z} wn~l + A2 (z) wn~2 + . . . = 0,

where h0(z) is the least common multiple of all the denominators of the
functions Ft. The preceding results therefore lead to the more limited
theorem :

When a function w is determined by an algebraical equation the coefficients
of which are holomorphic functions of z, then each of the zeros of the coefficient
of the highest power of w is an infinity of some of (and it may be of all) the
branches of the function w, each such infinity being of finite order. The point
z= oo may also be an infinity of the function w ; the order of that infinity is
finite or infinite according as z = oo is an accidental or an essential singularity
of any of the coefficients.

It will be noticed that no precise determination of the forms of the
branches w at an infinity has been made. The determination has, however,
only been deferred : the infinities of the branches for a singularity of the
coefficients are usually associated with a branch-point of the function and
therefore the relations of the branches at such a point will be of a general
character independent of the fact that the point is an infinity.

If, however, in any case a singularity of a coefficient should prove to be,
not a branch-point of w but only a regular singularity, then in the vicinity of
that point the branch of w is a uniform function. A necessary (but not suffi
cient) condition for uniformity is that (mn_r — mn_s) -7- (s — r) be an integer.

Note. The preceding method can be applied to determine the leading
terms of the branches in the vicinity of a point a which is an ordinary point
for each of the coefficients F.

168 BRANCH-POINTS [97.

97. There remains therefore the consideration of the branch-points of a
function determined by an algebraical equation.

The characteristic property of a branch-point is the equality of branches
of the function for the associated value of the variable, coupled with the
interchange of some of (or all) the equal branches after description by the
variable of a small contour enclosing the point.

So far as concerns the first part, the general indication of the form of the
values has already (§ 93) been given. The points, for which values of w
determined as a function of z by the equation

f(w, z) =J0

are equal, are determined by the solution of this equation treated simul
taneously with

df(w, z) = Q.
dw

and when a point z is thus determined the corresponding values of w, which
are equal there, are obtained by substituting that value of z and taking M,

the greatest common measure of / and -J- . The factors of M then lead to

the value or the values of w at the point ; the index m of a linear factor
gives at the point the multiplicity of the value which it determines, and
shews that m + 1 values of w have a common value there, though they are
distinct at infinitesimal distances from the point. If m = 1 for any factor,
the corresponding value of w is an isolated value and determines a branch
that is uniform at the point.

Let z = a, w = a be a value of z and a value of w thus obtained ; and
suppose that m is the number of values of w that are equal to one another.
The point z = a is not a branch-point unless some interchange among the
in values of w is effected by a small circuit round a ; and it is therefore
necessary to investigate the values of the branches* in the vicinity of z — a.

Let w = a. + w', z = a + z' ; then we have

that is, on the supposition that f(w, z) has been freed from fractions,

/(a, a) + SS^rXV = 0,

r, s

so that, since a is a value of w corresponding to the value a of z, we have
w' and / connected by the relation

* The following investigations are founded on the researches of Puiseux on algebraic
functions; they are contained in two memoirs, Liouville, lre Ser., t. xv, (1850), pp. 365 — 480, ib.,
t. xvi, (1851), pp. 228—240. See also the chapters on algebraic functions, pp. 19 — 76, in the
second edition of Briot and Bouquet's Theorie des fonctions elliptiques.

97.] OF ALGEBRAICAL FUNCTIONS 169

When / is 0, the zero value of w' must occur m times, since a is a root
m times repeated; hence there are terms in the foregoing equation inde

pendent of z, and the term of lowest index among them is w'm. Also when
w ' = 0, z' — 0 is a possible root ; hence there must be a term or terms
independent of w' in the equation.

First, suppose that the lowest power of z among the terms independent
of w' is the first. The equation has the form

Az' + higher powers of z'
+ Biu' + higher powers of w'
+ terms involving z' and w' = 0,

O-/* / \

where A is the value of - ' — - for w = a, z = a. Let z'=%m, w' = v%: the

02

last form changes to

(A + Bvm) £m + terms with £m+1 as a factor = 0 ;
and therefore A + Bvm + terms involving £= 0.

Hence in the immediate vicinity of z = a, that is, of £ =0, we have

A + Bv™ = 0.

Neither A nor B is zero, so that all the m values of v are finite. Let them
be vl}..., vm, so arranged that their arguments increase by 2-Tr/Tn through
the succession. The corresponding values of w' are

for i = l, ..., m. Now a ^-circuit round a, that is, a /-circuit round its
origin, increases the argument of z' by 2?r ; hence after such a circuit we

1_ 27Tt !_

have the new value of w{ as ViZ/m em, that is, it is vi+1z'm which is the value
of w'i+l. Hence the set of values w\, «/.,,..., w'm form a complete set of
interchangeable values in their cyclical succession ; all the m values, which
are equal at a, form a single cycle and the point is a branch-point.

Next, suppose that the lowest power of z among the terms independent
of w is z' , where I > 1. The equation now has the form
0 = Az' + higher powers of z'
+ Bw' + higher powers of w'

Arsz'V

r=l s=l

where in the last summation r and s are not zero and in every term either
(i), r is equal to or greater than I or (ii), s is equal to or greater than m
or (iii), both (i) and (ii) are satisfied. As only terms of the lowest orders

170

BRANCH-POINTS

[97.

need be retained for the present purpose, which is the derivation of the first
term of w' in its expansion in powers of z', we may use the foregoing equation
in the form

, l-lm-l

A/ + 2 2 A,

r=l s=l

,r ,s , -p. ,m _

jf w + Bw = 0.

To obtain this first term we proceed in a manner similar to that in § 96 *.
Points A0,..., Am are taken in a plane
referred to rectangular axes having as co
ordinates 0, £;...; s, r;...; m, 0 respectively.
A line is taken through Am and is made to
turn round Am from the position AmO until
it first meets one of the other points ; then
round the last point which lies in this
direction, say round Aj, until it first meets
another ; and so on.

Any line through At (the point si} rt) is
of the form

y - Ti = - \ (x - s^.

The intercept on the axis of /-indices is \Si + Ti, that is, the order of the
term involving Ars for a substitution w' oc / . The perpendicular from the
origin for a line through AI and Aj is less than for any parallel line through
other points with the same inclination ; and, as this perpendicular is

Fig. 21.

it follows that, for the particular substitution w' oc z' , the terms corresponding
to the points lying on the line with coordinate X are the terms of lowest
order and consequently they are the terms which give the initial terms for
the associated set of quantities w'.

Evidently, from the indices retained in the equation, the quantities X
for the various pieces of the broken line from Am to A0 are positive and
finite.

Consider the first piece, from Am to Aj say ; then taking the value of X for

that piece as fa, so that we write v^z'*1 as the first term of w', we have as the
set of terms involving the lowest indices

J? /"* i ^ ^ A fl* fi I A fl*i , ^J

Sj being the smallest value of s retained ; and then

so that

/*! =

m — s

* Reference in this connection may be made to Chrystal's Algebra, ch. xxx., with great
advantage, as well as the authorities quoted on p. 168, note.

GROUPING OF BRANCHES 171

Let p/q be the equivalent value of ^ as the fraction in its lowest terms ; and

p

write / = (?. Then w' = vlz'i = vtf ; all the terms except the above group

are of order > mp and therefore the equation leads after division by %mPtfi to

Bv^-'i + ^Aravf-*i + Arfj = 0,

an equation which determines m — Sj values for vl, and therefore the initial
terms of m — Sj of the w-branches.

Consider now the second piece, from Aj to At say ; then taking the value

of A, for that piece as fa, so that we write v.2z'^ as the first term of w', we
have as the set of terms involving the lowest indices for this value of /*2

A fri /s.- xr"O A iv Is fl"i tsi

Arfz' Jw ' + E&A.rjt w' + Ar.sz' *w \
where S{ is the smallest value of s retained. Then

Sjfr + Tj = tyig + r

Proceeding exactly as before, we find

as the equation determining Sj-Si values for v2 and therefore the initial
terms of Sj — st of the w-branches.

And so on, until all the pieces of the line are used ; the initial terms of
all the w-branches are thus far determined in groups connected with the
various pieces of the line A^Ai^.A,. By means of these initial terms,
the m-branches can be arranged for their interchanges, by the description of
a small circuit round the branch-point, according to the following theorem :—

Each group can be resolved into systems, the members of each of which are
cyclically interchangeable.

It will be sufficient to prove this theorem for a single group, say the
group determined by the first piece of broken line: the argument is
general.

Since - is the equivalent of — ^— and of T} . and since s, < s, we have
V m — s m — Sj

m-s = kq, m-Sj^kjq, kj>k;
and then the equation which determines ^ is

Sv&v + 2^r,,Vl <*,-*> 1 4 ArjSj = 0,

that is, an equation of degree k} in vj as its variable. Let U be any root of
it ; then the corresponding values of vl are the values of U*. Suppose these
q values to be arranged so that the arguments increase by 27r^, which is

possible, because p is prime to q. Then the q values of w' being the values
of v^Vi are

P. P P

172 GROUPING OF BRANCHES [97.

where vla is that value of Ifi which has — — for its argument. A circuit

round the /-origin evidently increases the argument of any one of these
w'-values by Zrrp/q, that is, it changes it into the value next in the succession;
and so the set of q values is a system the members of which are cyclically
interchangeable.

This holds for each value of U derived from the above equation ; so that
the whole set of m — Sj branches are resolved into kj systems, each containing
q members with the assigned properties.

It is assumed that the above equation of order kj in vj has its roots unequal.
If, however, it should have equal roots, it must be discussed ab initio by a
method similar to that for the general equation; as the order kj (being a
factor of m — Sj) is less than m, the discussion will be shorter and simpler,
and will ultimately depend on equations with unequal roots as in the case
above supposed.

It may happen that some of the quantities /j, are integers, so that the
corresponding integers q are unity : a number of the branches would then be
uniform at the point.

It thus appears that z = a is a branch-point and that, under the present
circumstances, the m branches of the function can be arranged in systems,
the members of each one of which are cyclically interchangeable.

Lastly, it has been tacitly assumed in what precedes that the common
value of w for the branch-point is finite. If it be infinite, this infinite value
can, by § 95, arise only out of singularities of the coefficients of the equation :
and there is therefore a reversion to the discussion of §§ 95, 96. The dis
tribution of the various branches into cyclical systems can be carried out
exactly as above.

Another method of proceeding for these infinities would be to take
ww' = \, z= c + z' ; but this method has no substantial advantage over the
earlier one and, indeed, it is easy to see that there is no substantial
difference between them.

Ex. 1. As an example, consider the function determined by the equation

The equation determining the values of z which give equal roots for w is

82 (2 -1)2 = 4(3 -I)3
so that the values are z=l (repeated) and z= — 1.

When z=l, then w=0, occurring thrice; and, if 2 = 1+2' then

8W/3W,

that is, w'^^z13.

The three values are branches of one system in cyclical order for a circuit round z=\.

97.] EXAMPLES 173

When z = — 1, the equation for w is

that is, (w

so that w=\ or w= — £, occurring twice.

For the former of these we easily find that, for s= — l-\-z', the value of w is
l-hfs'-f ...... , an isolated branch as is to be expected, for the value 1 is not

repeated.

For the latter we take w—— \ + w' and find

so that the two branches are

and they are cyclically interchangeable for a small circuit round z= - 1.

These are the finite values of w at branch-points. For the infinities of w, which may
arise in connection with the singularities of the coefficients, we take the zeros of the
coefficient of the highest power of w in the integral equation, viz., 2 = 0, which is thus the
only infinity of w. To find its order we take w=z~nf (z)—yz~n + ...... , where y is a

constant and f (z) is finite for 2 = 0; and then we have

8zl~3n

J. "~

Thus l-3n=-n,

provided both of them be negative; the equality gives n = \ and satisfies the condition.
And 8y3= - 3y. Of these values one is zero, and gives a branch of the function without
an infinity; the other two are ±^V-f and they give the initial term of the two
branches of w, which have an infinity of order -^ at the origin and are cyclically
interchangeable for a small circuit round it. The three values of w for infinitesimal
values of z are

3 . _i 1 7 /3 . l 4 275
" - 81 '-1944

3 • -

M + — /?&*— 1*4. 215 /3-f_jL2_

6 18 V 8 81 1944 V 8 729 z

_ _i A As
w3--g + gj2+— 2 +

The first two of these form the system for the branch-point at the origin, which is neither
an infinity nor a critical point for the third branch of the function.

Ex. 2. Obtain the branch-points of the functions which are defined by the following
equations, and determine the cyclical systems at the branch-points :
(i) w*
(ii) w
(iii) w
(iv) iff

44
(v) vfi - (1 - a2) 104 _ _ Z2 (! _ 22)4 = 0- (Briot and Bouquet.)

Also discuss the branches, in the vicinity of 2 = 0 and of 2=00, of the functions defined
by the following equations :

(vi) aw7 + bu£z + cutz* + dwW + ewz1 +fz9 + gv£ + hw*£ + kzw = 0 ;
(vii) wmzn+wn+zm = Q.

174 SIMPLE BRANCH-POINTS [98.

98. There is one case of considerable importance which, though limited
in character, is made the basis of Clebsch and Gordan's investigations* in the
theory of Abelian functions — the results being, of course, restricted by the
initial limitations. It is assumed that all the branch-points are simple, that
is, are such that only one pair of branches of w are interchanged by a circuit
of the variable round the point ; and it is assumed that the equation /= 0 is
algebraical not merely in w but also in z. The equation f = 0 can then be
regarded as the generalised form of the equation of a curve of the nth order,
the generalisation consisting in replacing the usual coordinates by complex
variables; and it is further assumed, in order to simplify the analysis, that all
the multiple points on the curve are (real or imaginary) double-points. But,
even with the limitations, the results are of great value : and it is therefore
desirable to establish the results that belong to the present section of the
subject.

We assume, therefore, that the branch-points are such that only one
pair of branches of w are interchanged by a small closed circuit round any
one of the points. The branch-points are among the values of z determined
by the equations

z) A
>

When /=0 has the most general form consistent with the assigned
limitations, f (w, z) is of the ?ith degree in z ; the values of z are determined
by the eliminant of the two equations which is of degree n(n — 1), and there
are, therefore, n(n — Y) values of z which must be examined.

First, suppose that J \, ' — ' does not vanish for a value of z, thus

oz

obtained, and the corresponding value of w : then we have the first case
in the preceding investigation. And, on the hypothesis adopted in the
present instance, m = 2 ; so that each such point z is a branch-point.

Next, suppose that — ^ - vanishes for some of the n(n — 1) values of z ;

the value of m is still 2, owing to the hypothesis. The case will now be still

d'2f (w z}
further limited by assuming that ^ .2 does not vanish for the value of z

and the corresponding value of w ; and thus in the vicinity of z = a, w = a we
have an equation

0 = Az- + 2Bz'w' + Cw'2 -f terms of the third degree + ...... ,

where A, B, C are the values of ^ , =-£- , «~ f°r z — a> w=a.

oz1 dzdw ow2

If B2 AC, this equation leads to the solution

C'w + Bz oc uniform function of z.

* Clebsch und Gordan, Theorie der AbeVschen Functionen, (Leipzig, Teubner, 1866).

98.] SIMPLE BRANCH-POINTS 175

The point z = a, w = a is not a branch-point ; the values of w, equal at the
point, are functionally distinct. Moreover, such a point z occurs doubly in
the eliminant; so that, if there be B such points, they account for 28 in
the eliminant of degree n (n — 1) ; and therefore, on their score, the number
n (n — 1) must be diminished by '28. The case is, reverting to the genera
lisation of the geometry, that of a double point where the tangents are
not coincident.

If, however, B2 = AC, the equation leads to the solution

Cw' + Bz' = Lz'^ + Mz'* + Nz'* +

The point z = a, w = a is a point where the two values of z interchange.
Now such a point z occurs triply in the eliminant ; so that, if there be K
such points, they account for SK of the degree of the equation. Each of
them provides only one branch-point, and the aggregate therefore provides K
branch-points ; hence, in counting the branch-points of this type as derived
through the eliminant, its degree must be diminished by 2/c. The case is,
reverting to the generalisation of the geometry, that of a double point (real
or imaginary) where the tangents are coincident.

It is assumed that all the n(n— 1) points z are accounted for under
the three classes considered. Hence the number of branch-points of the
equation is

£l = n (n - 1) - 28 - 2«,

where n is the degree of the equation, B is the number of double points
(in the generalised geometrical sense) at which tangents to the curve do not
coincide, and K is the number of double points at which tangents to the
curve do coincide.

And at each of these branch-points, II in number, two branches of the
function are equal and, for a small circuit round it, interchange.

99. The following theorem is a combined converse of many of the
theorems which have been proved :

A function w, which has n (and only ?/) values for each value of z, and
which has a finite number of infinities and of branch-points in any part of the
plane, is a root of an equation in w of degree n, the coefficients of which are
uniform functions of z in that part of the plane.

We shall first prove that every integral symmetric function of the n
values is a uniform function in the part of the plane under consideration.

n

Let Sk denote 2, w£, where k is a positive integer. At an ordinary point

i-\

of the plane, Sk is evidently a one-valued function and that value is finite ;
Sk is continuous ; and therefore the function Sk is uniform in the immediate
vicinity of an ordinary point of the plane.

176 FUNCTIONS POSSESSING [99.

For a point a, which is a branch-point of the function w, we know that
the branches can be arranged in cyclical systems. Let w1,..., w^ be such a
system. Then these branches interchange in cyclical order for a description
of a small circuit round a ; and, if z — a = Z*, it is known (§ 93) that, in the
vicinity of Z = 0, a branch w is a, uniform function of Z, say

Therefore wk = Gk ) + Pk (Z)

\£il

in the vicinity of Z = 0 ; say

w* = Ak + 2Bk>mZ-™ + 2 Ck>mZ™.

m=l m=l

Now the other branches of the function which are equal at a are derivable
from any one of them by taking the successive values which that one
acquires as the variable describes successive circuits round a. A circuit
of w round a changes the argument of z — a, by 27r. and therefore gives Z
reproduced but multiplied by a factor which is a primitive /xth root of unity,
say by a factor a ; a second circuit will reproduce Z with a factor a2 ; and so
on. Hence

wf = Ak+2 Bk>m a— Z-™ +ZCk>m a- #»

wrk = Ak+2 Bk>m a-™ Z~m + 2 Ck,m a™ Zm,

m=l »»=!

and therefore
I*

wrk = pAk + 2 Bkm - + ar + cr + . . . + cr't

r=l m = 1

+ 2 flto* Zm (1 + «m + a2"* + • • • + a""*-™).

OT = 1

Now, since a is a primitive /*th root of unity,

1 +as + «2S+ ... + as('x-1)

is zero for all integral values of s which are not integral multiples of p,, and it
is yu, for those values of s which are integral values of jj, ; hence

- £

B'k> i(z - a)"1 + B'k^(z - a)~2 + B'kt3 (z - a)

.
Hence the point z = a may be a singularity of 2 wrk but it is not a branch-

r=l

99.] A FINITE NUMBER OF BRANCHES 177

point of the function ; and therefore in the immediate vicinity of z — a the

*i
quantity X wrk is a uniform function.

r=l

The point a is an essential singularity of this uniform function, if the
order of the infinity of w at a be infinite : it is an accidental singularity, if
that order be a finite integer.

This result is evidently valid for all the cyclical systems at a, as well as
for the individual branches which may happen to be one-valued at a. Hence

(U.

Sk, being the sum of sums of the form 2) wrk each of which is a uniform

r=l

function of z in the vicinity of a, is itself a uniform function of z in that
vicinity. Also a is an essential singularity of Sk, if the order of the infinity at
z = a for any one of the branches of w be infinite ; and it is an accidental
singularity of Sk> if the order of the infinity at z = a for all the branches of w
be finite. Lastly, it is an ordinary point of Sk, if there be no branch of w for
which it is an infinity. Similarly for each of the branch-points.

Again, let c be a regular singularity of any one (or more) of the branches
of w ; then c is a regular singularity of every power of each of those branches,
the singularities being simultaneously accidental or simultaneously essential.
Hence c is a singularity of 8k : and therefore in the vicinity of c, $& is a
uniform function, having c for an accidental singularity if it be so for each of
the branches w affected by it, and having c for an essential singularity if it be
so for any one of the branches w.

It thus appears that in the part of the plane under consideration the
function 8k is one-valued ; and it is continuous and finite, except at certain
isolated points each of which is a singularity. It is therefore a uniform
function in that part of the plane ; and the singularity of the function at any
point is essential, if the order of the infinity for any one of the branches w at
that point be infinite, but it is accidental, if the order of the infinity for all the
branches w there be finite. And the number of these singularities is finite,
being not greater than the combined number of the infinities of the function
w, whether regular singularities or branch-points.

Since the sums of the kth powers for all positive values of the integer k
are uniform functions and since any integral symmetric function of the n
values is a rational integral algebraical function of the sums of the powers, it
follows that any integral symmetric function of the n values is a uniform
function of z in the part of the plane under consideration ; and every infinity
of a branch w leads to a singularity of the symmetric function, which is
essential or accidental according as the orders of infinity of the various
branches are not all finite or are all finite.

F. 12

178 FUNCTIONS POSSESSING [99.

Since w has n (and only n) values wlt... ,wn for each value of z, the
equation which determines w is

(W - Wj) (W-W2) ... (W- Wn) = 0.

The coefficients of the various powers of w are symmetric functions of the
branches wl , . . . , wn; and therefore they are uniform functions of z in the
part of the plane under consideration. They possess a finite number of
singularities, which are accidental or essential according to the character of
the infinities of the branches at the same points.

COROLLARY. If all the infinities of the branches in the finite part of the
whole plane be of finite order, then the finite singularities of all the coefficients
of the powers of w in the equation satisfied by w are all accidental ; and the
coefficients themselves then take the form of a quotient of an integral uniform
function (which may be either transcendental or algebraical, in the sense of
§ 47) by another function of a similar character.

If z = oc be an essential singularity for at least one of the coefficients,
through being an infinity of unlimited order for a branch of w, then one
or both of the functions in the quotient-form of one at least of the coefficients
must be transcendental.

If z = oo be an accidental singularity or an ordinary point for all the
coefficients, through being either an infinity of finite order or an ordinary
point for the branches of w, then all the functions which occur in all the
coefficients are rational, algebraical expressions. When the equation is
multiplied throughout by the least common multiple of the denominators
of the coefficients, it takes the form

wnh0 (z) + wn~* A, (z) + . . . + w hn_, (z} + hn (z) = 0,

where the functions h0(z), h^(z\ ..., hn(z) are rational, integral, algebraical
functions of z, in the sense of § 47.

A knowledge of the number of infinities of w gives an upper limit of the
degree of the equation in z in the last form. Thus, let at be a regular
singularity of the function ; and let Oi, fa, ji, ... be the orders of the infinities
of the branches at at- ; then

w^w-i ... wn(z — at )A',
where \ denotes Oi + fti + % + ..., is finite (but not zero) for z = at.

Let Ci be a branch-point, which is an infinity; and let p, branches w form a

ft
system for ct-, such that w(z — Cf)^ is finite (but not zero) at the point; then

w:w2 ... Wp (z — Q) '
is finite (but not zero) at the point, and therefore also

99.] A FINITE NUMBER OF BRANCHES 179

is finite, where Qit (/>;, ^i, ... are numbers belonging to the various systems;
or, if ei denote 0; + $f + tyi + . . . , then

Wl...Wn(z- Ci)6i

is finite for z = C;. Similarly for other symmetric functions of w.

Hence, if «j, a2, ... be the regular singularities with numbers X1; X2, ...
defined as above, and if c^ c2, ... be the branch -points, that are also infinities,
with numbers e1; e2, ... defined as above, then the product

(w-Wj) ...... (w-wn) n 0-af)A< n 0-Ci)e<

i=l 1=1

is finite at all the points ai and at all the points c;. The points a and the
points c are the only points in the finite part of the plane that can make the
product infinite : hence it is finite everywhere in the finite part of the plane,
and it is therefore an integral function of z.

Lastly, let p be the number for z = oo corresponding to \i for af or to e^
for C;, so that for the coefficient of any power of w in (w — w^) ...(w — wn) the
greatest difference in degree between the numerator and the denominator is
p in favour of the excess of the former.

Then the preceding product is of order

which is therefore the order of the equation in z when it is expressed in a
holomorphic form.

12—2

CHAPTER IX.

PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS IN GENERAL.

100. INSTANCES have already occurred in which the value of a function
of z is not dependent solely upon the value of z but depends also on the
course of variation by which z obtains that value ; for example, integrals of
uniform functions, and multiform functions. And it may be expected that,
a fortiori, the value of an integral connected with a multiform function will
depend upon the course of variation of the variable z. Now as integrals
which arise in this way through multiform functions and, generally, integrals
connected with differential equations are a fruitful source of new functions,
it is desirable that the effects on the value of an integral caused by variations
of a £-path be assigned so that, within the limits of algebraic possibility, the
expression of the integral may be made completely determinate.

There are two methods which, more easily than others, secure this result ;
one of them is substantially due to Cauchy, the other to Riemann.

The consideration of Riemann's method, both for multiform functions and
for integrals of such functions, will be undertaken later, in Chapters XV.,
XVI. Cauchy's method has already been used in preceding sections relating
to uniform functions, and it can be extended to multiform functions. Its
characteristic feature is the isolation of critical points, whether regular
singularities or branch-points, by means of small curves each containing one
and only one critical point.

Over the rest of the plane the variable z ranges freely and, under certain
conditions, any path of variation of z from one point to another can, as will
be proved immediately, be deformed without causing any change in the
value of the integral, provided that the path does not meet any of the small
curves in the course of the deformation. Further, from a knowledge of the
relation of any point thus isolated to the function, it is possible to calculate
the change caused by a deformation of the £-path over such a point; and
thus, for defined deformations, the value of the integral can be assigned
precisely.

100.] INTEGRAL OF A BRANCH 181

The properties proved in Chapter II. are useful in the consideration of
the integrals of uniform functions ; it is now necessary to establish the
propositions which give the effects of deformation of path on the integrals
of multiform function. The most important of these propositions is the
following : —

fb
If w be a multiform function, the value of I wdz, taken between two

J a

ordinary points, is unaltered for a deformation of the path, provided that the
initial branch of w be the same and that no branch-point or infinity be crossed
in the deformation.

Consider two paths acb, adb, (fig. 16, p. 152), satisfying the conditions
specified in the proposition. Then in the area between them the branch w
has no infinity and no point of discontinuity ; and there is no branch-point
in that area. Hence, by § 90, Corollary VI., the branch w is a uniform
monogenic function for that area; it is continuous and finite everywhere
within it and, by the same Corollary, we may treat w as a uniform, mono
genic, finite and continuous function. Hence, by § 17, we have

rb ra

(c) I wdz + (d) wdz = 0,

J a J b

the first integral being taken along acb and the second along bda; and
therefore

rb ra rb

(c) wdz = — (d}\ wdz = (d) \ wdz,

Jo, J b J a

shewing that the values of the integral along the two paths are the same
under the specified conditions.

It is evident that, if some critical point be crossed in the deformation,
the branch w cannot be declared uniform and finite in the area and the
theorem of § 17 cannot then be applied.

COROLLARY I. The integral round a closed curve containing no critical
point is zero.

COROLLARY II. A curve round a branch-point, containing no other
critical point of the function, can be deformed into a loop
without altering the value of fwdz ; for the deformation
satisfies the condition of the proposition. Hence, when
the value of the integral for the loop is known, the
value of the integral is known for the curve.

COROLLARY III. From the proposition it is possible
to infer conditions, under which the integral fwdz round
the whole of any curve remains unchanged, when the whole
curve is deformed, without leaving an infinitesimal arc
common as in Corollary II.

182 INTEGRATION [100.

Let GDC', ABA' be the curves: join two consecutive points A A' to two
consecutive points (7(7. Then if the area CABA'C'DG
enclose no critical point of the function w, the value of
jwdz along CDC' is by the proposition the same as its
value along CABA'C'. The latter is made up of the
value along CA, the value along ABA', and the value
along AC', say

rA r rC'

I wdz + I wdz + w'dz, v.

Jc JB JA Ǥ.

where w' is the changed value of w consequent on the description of a simple
curve reducible to B (§ 90, Cor. II.).

Now since w is finite everywhere, the difference between the values of w
at A and at A' consequent on the description of ABA is finite : hence as
A A is infinitesimal the value of jwdz necessary to complete the value for
the whole curve B is infinitesimal and therefore the complete value can be

taken as the foregoing integral wdz. Similarly for the complete value

J B

along the curve D : and therefore the difference of the integrals round B and
round D is

rA rC'

I wdz + I w'dz,

J C J A'

rA

say (w — w') dz.

J c

In general this integral is not zero, so that the values of the integral
round B and round D are not equal to one another : and therefore the curve
D cannot be deformed into the curve B without affecting the value of jwdz
round the whole curve, even when the deformation does not cause the curve
to pass over a critical point of the function.

But in special cases it may vanish. The most important and, as a
matter of fact, the one of most frequent occurrence is that in which the
description of the curve B restores at A' the initial value of w at A. It
easily follows, by the use of § 90, Cor. II., that the description of D (as
suming that the area between B and D includes no critical point) restores
at C' the initial value of w at (7. In such a case, w = w' for corresponding
points on AC and A'C', and the integral, which expresses the difference, is
zero: the value of the integral for the curve B is then the same as that for D.
Hence we have the proposition : —

If a curve be such that the description of it by the independent variable
restores the initial value of a multiform function w, then the value of jwdz
taken round the curve is unaltered when the curve is deformed into any other
curve, provided that no branch-point or point of discontinuity of w is crossed
in the course of deformation.

100.] OF MULTIFORM FUNCTIONS 183

This is the generalisation of the proposition of § 19 which has thus far
been used only for uniform functions.

Note. Two particular cases, which are very simple, may be mentioned
here : special examples will be given immediately.

The first is that in which the curve B, and therefore also D, encloses
no branch-point or infinity; the initial value of w is restored after a
description of either curve, and it is easy to see (by reducing B to a
point, as may be done) that the value of the integral is zero.

The second is that in which the curve encloses more than one branch
point, the enclosed branch-points being such that a circuit of all the loops,
into which (by Corollary V., § 90) the curve can be deformed, restores the
initial branch of w. This case is of especial importance when w is two-valued :
the curves then enclose an even number of branch-points.

101. It is important to know the value of the integral of a multiform
function round a small curve enclosing a branch-point.

Let c be a point at which TO branches of an algebraical function are equal
and interchange in a single cycle ; and let c, if an infinity, be of only finite
order, say k/m. Then in the vicinity of c, any of the branches w can be
expressed in the form

00 .«

w= 2 gs(z-c)m,

o — If

o — — K

where k is a finite integer.

The value of jwdz taken round a small curve enclosing c is the sum of
the integrals

the value of which, taken once round the curve and beginning at a point zly is

TO + S

where a is a primitive mth root of unity, provided TO + s is not zero. If then
s + m be positive, the value is zero in the limit when the curve is infini
tesimal : if TO + s be negative, the value is oo in the limit.

But, if m + s be zero, the value is Z7rigs.

Hence we have the proposition: If, in the vicinity of a branch-point c,
where m branches w are equal to one another and interchange cyclically, the
expression of one of the branches be

184 MULTIPLICITY OF VALUE [101.

then jwdz, taken once round a small curve enclosing c, is zero, if k<m; is
infinite, if k> m ; and is ^irig^ , if k = m.

It is easy to see that, if the integral be taken m times round the small
curve enclosing c, then the value of the integral is 2m7rigm when k is greater
than in, so that the integral vanishes unless there be a term involving (z — c)"1
in the expansion of a branch w in the vicinity of the point. The reason that
the integral, which can furnish an infinite value for a single circuit, ceases to

_*

do so for m circuits, is that the quantity (^ — c) m, which becomes indefi
nitely great in the limit, is multiplied for a single circuit by a*— 1, for a
second circuit by a2A — aA, and so on, and for the mth circuit by awA — a(w~1)A,
the sum of all of which coefficients is zero.

Ex. The integral \{(z - a) (z - b) ... (z -f)}~* dz taken round an indefinitely small curve
enclosing a is zero, provided no one of the quantities b, ... ,/ is equal to a.

102. Some illustrations have already been given in Chapter II., but
they relate solely to definite, not to indefinite, integrals of uniform
functions. The whole theory will not be considered at this stage ; we shall
merely give some additional illustrations, which will shew how the method
can be applied to indefinite integrals of uniform functions and to integrals
of multiform functions, and which will also form a simple and convenient
introduction to the theory of periodic functions of a single variable.

We shall first consider indefinite integrals of uniform functions.

f dz
Ex. 1. Consider the integral I — , and denote* it by/ (z}.

The function to be integrated is uniform, and it has an accidental singularity of the first
order at the origin, which is its only singularity. The value of \z~l dz taken positively
along a small curve round the origin, say round a circle with the origin as centre, is 2n-i •
but the value of the integral is zero when taken along any closed curve which does not
include the origin.

Taking z = l as the lower limit of the integral, and any point z as the upper limit, we
consider the possible paths from 1 to z. Any path from 1 to z can be deformed, without
crossing the origin, into a path which circumscribes the origin positively some number of
times, say m^, and negatively some number of times, say »i2, all in any order, and then leads
in a straight line from 1 to z. For this path the value of the integral is equal to

I — ,
J 1 z

that is, to 2mni+ I — ,

Ji z

where m is an integer, and in the last integral the variation of z is along a straight
line from 1 to z. Let the last integral be denoted by u ; then

* See Chrystal, ii, pp. 266 — 272, for the elementary properties of the function and its inverse,
when the variable is complex.

102.] OF INTEGRALS 185

and therefore, inverting the function and denoting/"1 by <j>, we have

Hence the general integral is a function of z with an infinite number of values ; and z is a
periodic function of the integral, the period being 2n-z.

Ex. 2. Consider the function / - - ^ > and again denote it by / (z).

The one- valued function to be integrated has two accidental singularities + i, each of
the first order. The value of the integral taken positively along a small curve round i is
TT, and along a small curve round — i is — n.

We take the origin 0 as the lower limit and any point z as the upper limit. Any path
from 0 to z can be deformed, without crossing either of the singularities and therefore
without changing the value of the integral, into

(i) any numbers of positive (ml5 w?2) an(* of negative (nz/, m2') circuits round i and
round -i, and

(ii) a straight line from 0 to z.
Then we have

- TJ-) +WIJJ ( - IT) + m.2' {_(-„•)}+ /* .

J o

, z
= nir+

where ?i is an integer and the integral on the right-hand side is taken along a straight line
from 0 to z.

Inverting the function and denoting/"1 by tp, we have

The integral, as before, is a function of z with an infinite number of values ; and z is a
periodic function of the integral, the period being TT.

103. Before passing to the integrals of multiform functions, it is con
venient to consider the method in which Hermite* discusses the multiplicity
in value of a definite integral of a uniform function.

Taking a simple case, let <£> (X) = \

J Q 1 + Z

and introduce a new variable t such that Z—zt\ then

zdt

When the path of t is assigned, the integral is definite, finite and unique in
value for all points of the plane except for those for which 1 + zt = 0 ; and,
according to the path of variation of t from 0 to 1, there will be a 0-curve
which is a curve of discontinuity for the subject of integration. Suppose the
path of t to be the straight line from 0 to 1 ; then the curve of discontinuity

* Crelle, t. xci, (1881), pp. 62—77; Cours a la Faculte des Sciences, 46me 6d. (1891), pp.
76—79, 154—164, and elsewhere.

186 HERMITE'S [103.

is the axis of x between — 1 and — oo . In this curve let any point - £ be
taken where £ > 1 ; and consider a point z1 — -^ + ie and a point z2 = — £ — ie,
respectively on the positive and the negative sides of the axis of x, both
being ultimately taken as infinitesimally near the point — £. Then

dt= (

Let e become infinitesimal ; then, when t is infinite, we have

tan

for e is positive ; and, when t is unity, we have

tan"1 ----- = — |TT,

for £ is > 1. Hence <£ (^) — <£ (^2)

The part of the axis of x from - 1 to - oo is therefore a line of discon
tinuity in value of <j> (z), such that there is a sudden change in passing from
one edge of it to the other. If the plane be cut along this line so that
it cannot be crossed by the variable which may not pass out of the plane,
then the integral is everywhere finite and uniform in the modified surface.
If the plane be not cut along the line, it is evident that a single passage
across the line from one edge to the other makes a difference of 2?ri in the
value, and consequently any number of passages across will give rise to the
multiplicity in value of the integral.

Such a line is called a section* by Hermite, after Riemann who, in a
different manner, introduces these lines of singularity into his method of
representing the variable on surfaces "f*.

When we take the general integral of a uniform function of Z and make
the substitution Z = zt, the integral that arises for consideration is of the form

We shall suppose that the path of variation of t is the axis of real quantities :
and the subject of integration will be taken to be a general function of t and
z, without special regard to its derivation from a uniform function of Z.

* Coupure; see Crelle, t. xci, p. 62. t See Chapter XV.

103.] SECTIONS 187

It is easy, after the special example, to see that ^ is a continuous function
of z in any space that does not include a ^-point which, for values of t included
within the range of integration, would satisfy the equation.

G (t, z) = 0.

But in the vicinity of a ^-point, say £, corresponding to the value t = 6 in
the range of integration, there will be discontinuity in the subject of
integration and also, as will now be proved, in the value of the integral.

Let Z be the point £ and draw the curve through Z corresponding to
t = real constant ; let Nt be a point on the positive side and N2
a point on the negative side of this curve positively described,
both points being on the normal at Z ; and let
supposed small. Then for N! we have

X-L = g — e sin y, yl = ^-\-e cos y ,

Fig. 24.
so that z1 = £+16' (cosy + isiny),

where ty is the inclination of the tangent to the axis of real quantities. But,
if da- be an arc of the curve at Z,

da , • • i \ d% • dt] d£

for variations along the tangent at Z, that is,

i

da- . . 3

-j-- (cos y + i sin y ) = — -

Thus, since -j- may be taken as finite on the supposition that Z is an
ordinary point of the curve, we have

where e = e' -y- , P = -

Similarly z.2 = £ + ie -^r.

Hence <1> (^) = I --i-i— *£ ^

w/ n_^J_w/ m _

1*.

188 HERMITE'S [103.

with a similar expression for <& (z2) ; and therefore

F(t, 0 j- [G (t, ®}^-G (t, §)

'

The subject of integration is infinitesimal, except in the immediate vicinity
of t = 6 ; and there

powers of small quantities other than those retained being negligible. Let
the limiting values of t, that need be retained, be denoted by d + v and
d — p', then, after reduction, we have

edt

F(e,

in the limit when e is made infinitesimal.

Hence a line of discontinuity of the subject of integration is a section
for the integral ; and the preceding expression is the magnitude, by
numerical multiples of which the values of the integral differ*.

Ex. 1. Consider the integral

dZ

/

zdt
h

We have S ^ *' =^ = ^g = ^.

so that TT is the period for the above integral.
Ex. 2. Shew that the sections for the integral

ta sin z ,
2 '

* The memoir and the Cmirs d' 'Analyse of Hermite should be consulted for further develop
ments; and, in reference to the integral treated above, Jordan, Cours d' Analyse, t. iii, pp.
610 — 614, may be consulted with advantage. See also, generally, for functions defined by
definite integrals, Goursat, Acta Math., t. ii, (1883), pp. 1—70, and ib., t. v, (1884), pp. 97—
120; and Pochhammer, Math. Arm., t. xxxv, (1890), pp. 470—494, 495—526. Goursat also
discusses double integrals.

103.] SECTIONS 189

where a is positive and less than 1, are the straight lines x = (2k + l) TT, where k assumes all
integral values ; and that the period of the integral at any section at a distance 77 from the
axis of real quantities is 2?r cosh (arj). (Hermite.)

Ex. 3. Shew that the integral

o

where the real parts of /3 and y — /3 are positive, has the part of the axis of real quantities
between 1 and +00 for a section.

Shew also that the integral

i

rht }— (z P~I (~i - vy~'3~1n— }~a d

J 0

where the real parts of /3 and 1 - a are positive, has the part of the axis of real quantities
between 0 and 1 for a section : but that, in order to render <£ (z) a uniform function of z,
it is necessary to prevent the variable from crossing, not merely the section, but also the
part of the axis of real quantities between 1 and + <x> . (Goursat.)

(The latter line is called a section of the second kind.)

Ex. 4. Discuss generally the effect of changing the path of t on a section of the

integral ; and, in particular, obtain the section for I — „ when, after the substitution

jo 1 + «

Z=zt, the path of t is made a semi-circle on the line joining 0 and 1 as diameter.

Note. It is manifestly impossible to discuss all the important bearings of theorems and
principles, which arise from time to time in our subject ; we can do no more than mention
the subject of those definite integrals involving complex variables, which first occur as
solutions of the better-known linear differential equations of the second order.

Thus for the definite integral connected with the hypergeometric series, memoirs by
Jacobi* and Goursat t should be consulted ; for the definite integral connected with
Bessel's functions, memoirs by HankelJ and Weber § should be consulted ; and Heine's
J/andbuch der Kugelfunctionen for the definite integrals connected with Legendre's
functions.

104. We shall now consider integrals of multiform functions.

Ex. 1. To find the integral of a multiform function round one loop ; and round a
number of loops.

Let the function be

i

w={(z-al}(z-a.z}...(z- an)}»» ,

where m may be a negative or positive integer, and the quantities a are unequal to one
another ; and let the loop be from the origin round the point ax. Then, if / be the value
of the integral with an assigned initial branch w, we have

/a, f CO

wdz-\- I wdz + I awdz,
0 J c J a.

where a is e m and the middle integral is taken round the circle at a^ of infinitesimal radius.

* Crelle, t. Ivi, (1859), pp. 149 — 165 ; the memoir was not published until after his death,
t Sur Vequation differentielle lineaire qui admet pour integrate la serie hypergrometrique,
(These, Gauthier-Villars, Paris, 1881).

I Math. Ann., t. i, (1869), pp. 467—501.

§ Math. Ann., t. xxxvii, (1890), pp. 404—416.

190

EXAMPLES

[104.

But, since the limit of (z-ajw when z = a1 is zero, the middle integral vanishes by § 101 ;
and therefore

/"«i
«, = (! -a) I web,

Jo

where the integral may, if convenient, be considered as taken along the straight line from
0 to al .

(2)

(3)

Fig. 25.

Next, consider a circuit for an integral of w which (fig. 25) encloses two branch-points,
say «! and «2, but no others ; the circuit in (1) can be deformed into that in (2) or into
that in (3) as well as into other forms. Hence the integral round all the three circuits
must be the same. Beginning with the same branch as in the first case, we have

(1

/«!
wdz,
o

as the integral after the first loop in (2). And the branch with which the second loop
begins is aw, so that the integral described as in the second loop is

/«2
awdz;
0

and therefore, for the circuit as in (2), the integral is

Cat [ay

1= (1 - a) I wdz + a (1 - a) / wdz.
Jo Jo

Proceeding similarly with the integral for the circuit in (3), we find that its expression is

/a2 /"<*!

wdz + a (I -a) I wdz,
0 J 0

and these two values must be equal.

But the integrals denoted by the same symbols are not the same in the two cases ; the

function I * wdz is different in the second value of J from that in the first, for the deforma-

Jo

tion of path necessary to change from the one to the other passes over the branch-point az.
In fact, the equality of the two values of / really determines the value of the integral for
the loop Oal in (3).

And, in general, equations thus obtained by varied deformations do not give relations
among loop-integrals but define the values of those loop-integrals for the deformed paths.

We therefore take that deformation of the circuit into loops which gives the simplest
path. Usually the path is changed into a group of loops round the branch-points as they
occur, taken in order in a trigonometrically positive direction.

The value of the integral round a circuit, equivalent to any number of loops, is obvious.

Ex. 2. To find the value of $wdz, taken round a simple curve which includes all the
branch-points of w and all the infinities.

104.] OF PERIODICITY OF INTEGRALS 191

If z = oo be a branch-point or an infinity, then all the branch-points and all the
infinities of w lie on what is usually regarded as the exterior of the curve, or the curve
may in one sense be said to exclude all these points. The integral round the curve is then
the integral of a function round a curve, such that over the area included by it the
function is uniform, finite and continuous ; hence the integral is zero.

If 0 = 00 be neither a branch-point nor an infinity, the curve can be deformed until it is
a circle, centre the origin and of very great radius. If then the limit of zw, when \z is
infinitely great, be zero, the value of the integral again is zero, by II., § 24.

Another method of considering the integral, is to use Neumann's sphere for the
representation of the variable. Any simple closed curve divides the area of the sphere
into two parts ; when the curve is defined as above, one of those parts is such that the
function is uniform, finite and continuous throughout and therefore its integral round the
curve, regarded as the boundary of that part, is zero. (See Corollary III., § 90.)

Ex. 3. To find the general value of J(l-22)~*cfe. The function to be integrated is
two-valued: the two values interchange round each of the branch-points ±1, which are
the only branch-points of the function.

Let / be the value of the integral for a loop from the origin round +1, beginning with
the branch which has the value +1 at the origin ; and let /' be the corresponding value
for the loop from the origin round - 1, beginning with the same branch. Then, by Ex. 1,

/= 2 P (1 - z*T*dz, /' = 2 f"1 (1 - z2)"* dz

= -/,

the last equality being easily obtained by changing variables.

Now consider the integral when taken round a circle, centre the origin and of indefinitely
great radius R ; then by § 24, II., if the limit of zw for z= QO be k, the value of \wdz round

this circle is 2iri&. In the present case w = (l- 22)~^ so that the limit of zw is + ^ ; hence

J(l-22r^2 = 27T,

the integral being taken round the circle. But since a description of the circle restores the

initial value, it can be deformed into the two loops from 0 O'

to A and from 0 to A'. The value round the first is /; and ^ r > ^

the branch with which the second begins to be described has

the value — 1 at the origin, so that the consequent value round *1S- ^"-

the second is — /' ; hence

7-/' = 2»r*

and therefore

verifying the ordinary result that

when the integral is taken along a straight line.

To find the general value of u for any path of variation between 0 and z, we proceed as
follows. Let Q be any circuit which restores the initial branch of (l-z2)~^. Then by
§ 100, Corollary II., Q may be composed of

(i) a set of double circuits round + 1, say m',
(ii) a set of double circuits round - 1, say m",
and (iii) a set of circuits round + 1 and - 1 ;

* It is interesting to obtain this equation when O' is taken as the initial point, instead of 0.

192 EXAMPLES OF PERIODICITY [104.

and these may come in any order and each may be described in either direction. Now for
a double circuit positively described, the value of the integral for the first description is /
and for the second description, which begins with the branch —(1 — z2)~^, it is — /; hence
for the double circuit it is zero when positively described, and therefore it is zero also when
negatively described. Hence each of the TO' double circuits yields zero as its nett contribu
tion to the integral.

Similarly, each of the m" double circuits round - 1 yields zero as its nett contribution
to the integral.

For a circuit round + 1 and - 1 described positively, the value of the integral has just
been proved to be /-/', and therefore when described negatively it is /'-/. Hence if
there be n^ positive descriptions and n2 negative descriptions, the nett contribution of all
these circuits to the value of the integral is (n± — n^) (I - 1'), that is, 2nir where n is an
inteer.

Hence the complete value for the circuit Q i

Now any path from 0 to z can be resolved into a circuit Q, which restores the initial
branch of (1 — 22)~ , chosen to have the value
+ 1 at the origin, and either (i) a straight
line Oz ;

or (ii) the path OACz, viz., a loop round
+ 1 and the line Oz ;

or (iii) the path OA'Cz, viz., a loop round
- 1 and the line Oz.

Let u denote the value for the line Oz, so that

u= f* (!-#)-* dk.

J o

Hence, for case (i), the general value of the integral is

2W7T + U.

For the path OA Cz, the value is 7 for the loop OAC, and is ( — u) for the line Cz, the
negative sign occurring because, after the loop, the branch of the function for integra
tion along the line is —(1 — 22)~5 ; this value is I—u, that is, it is TT — U. Hence, for case
(ii), the value of the integral is

— U.

For the path OA'Cz, the value is similarly found to be - TT - u ; and therefore, for case (iii),
the value of the integral is

2?wr — ir-u.

If /(z) denote the general value of the integral, we have either

Or /(Z) = (2TO+1)7T-W,

where n and m are any integers, so that/ (z) is a function with two infinite series of values.
Lastly, if z = $($) be the inverse oif(z} = 6, then the relation between u and z given by

can be represented in the form
and

104.] OF INTEGRALS 193

both equations being necessary for the full representation. Evidently z is a simply -periodic
function of u, the period being 2?r ; and from the definition it is easily seen to be an odd
function.

Let y = (\ -z2)—x (u\ so that y is an even function of u ; from the consideration of the
various paths from 0 to 2, it is easy to prove that

Ex. 4. To find the general value of f{(l-j*)(l-IM)}~*dk It will be convenient
(following Jordan *) to regard this integral as a special case of

Z= \{(z -a)(z- b) (z -c}(z- d)}~* dz = \wdz.

The two-valued function to be integrated has a, 6, c, d (but not oo ) as the complete
system of branch-points ; and the two values interchange at each of them. We proceed as
in the last example, omitting mere re-statements of reasons there given that are applicable
also in the present example.

Any circuit Q, which restores an initial branch of w, can be made up of
(i) sets of double circuits round each of the branch-points,
and (ii) sets of circuits round any two of the branch -points.
The value of \wdz for a loop from the origin to a branch-point k (where k = a,b, c, or d) is

2 I wdz ;

J o
and this may be denoted by K, where K=A, B, C, or D.

The value of the integral for a double circuit round a branch-point is zero. Hence the
amount contributed to the value of the integral by all the sets in (i) as this part of
Q is zero.

The value of the integral for a circuit round a and b taken positively is A - B ; for one
round b and c is B- C ; for one round c and d is C-D; for one round a and c is A - C,
which is the sum of A - B and B-C; and similarly for circuits round a and d and round
b and d. There are therefore three distinct values, say A-B, B-C, C-D, the values
for circuits round a and b, b and c, c and d respectively ; the values for circuits round any
other pair can be expressed linearly in terms of these values. Suppose then that the part
of Q represented by (ii), when thus resolved, is the nett equivalent of the description of m'
circuits round a and b, of n' circuits round b and c, and of I' circuits round c and d. Then
the value of the integral contributed by this part of Q is

• which is therefore the whole value of the integral for Q.

But the values of A, £, C, D are not independent f. Let a circle with centre the origin
and very great radius be drawn ; then since the limit of zw for |s| = oo is zero and since
2= cc is not a branch-point, the value of \wdz round this circle is zero (Ex. 2). The circle
can be deformed into four loops round a, b, c, d respectively in order ; and therefore the
value of the integral is A - B + C- D, that is,

Hence the value of the integral for the circuit fl is

where m and n denote m' - 1' and n' - 1' respectively.
* Cours d' Analyse, t. ii, p. 343.

t For a purely analytical proof of the following relation, see Greenhill's Elliptic Functions
Chapter II.

F- 13

194 PERIODICITY [104.

Now any path from the origin to z can be resolved into Q, together with either
(i) a straight line from 0 to z,

or (ii) a loop round a and then a straight line to z.

It might appear that another resolution would be given by a combination of Q with, say, a
loop round b and then a straight line to z ; but it is resoluble into the second of the above
combinations. For at C, after the description of the loop B , introduce a double description
of the loop A, which adds nothing to the value of the integral and does not in the end
affect the branch of w at C ; then the new path can be regarded as made up of (a) the
circuit constituted by the loop round b and the first loop round a, (/3) the second loop round
a, which begins with the initial branch of w, followed by a straight path to z. Of these
(a) can be absorbed into G, and (/3) is the same as (ii) ; hence the path is not essentially
new. Similarly for the other points.

Let u denote the value of the integral with a straight path from 0 to z; then the
whole value of the integral for the combination of Q with (i) is of the form

For the combination of O with (ii), the value of the integral for the part (ii) of
the path is J, for the loop round a, +(-«), for the straight path which, owing to the
description of the loop round a, begins with - w ; hence the whole value of the integral is
of the form

Hence, if / (z) denote the general value of the integral, it has two systems of values, each
containing a doubly -infinite number of terms; and, if z = <j>(u) denote the inverse of
u = f (z\ we have

= 0 {m (A-B} + n(B-C)+A - u},

where m and n are any integers. Evidently z is a doubly-periodic function of u, with
periods A-B and B-C.

Ex. 5. The case of the foregoing integral which most frequently occurs is the elliptic
integral in the form used by Legendre and Jacobi, viz. :

u = J{(1 - z2) (1 - kW)}-*dz = \wdz,
where k is real. The branch-points of the function to be integrated are 1, -1, ^

and -L and the values of the integral for the corresponding loops from the origin are

A/

A
2 I wdz,

J o

r-i ri

2 I wdz— -2 I wdz,
Jo /•

I wdz,
''

and

Now the values for the loops are connected by the equation

* The value for a loop round b and then a straight line to z, just considered, is B - u

= -(A-B) +
being the value in the text with m changed to m - 1.

104.]

OF ELLIPTIC INTEGRALS

195

and so it will be convenient that, as all the points lie on the axis of real variables, we
arrange the order of the loops so that this relation is identically satisfied. Otherwise,
the relation will, after Ex. 1, be a definition of the paths of integration chosen for the
loops.

Among the methods of arrangement, which secure the identical satisfaction of the

Fig. 28.

relation, the two in the figure* are the simplest, the curved lines being taken straight in
the limit ; for, by the first arrangement when k < 1, we have

and, by the second when £ > 1, we have

both of which are identically satisfied. We may therefore take either of them ; let the
former be adopted.

The periods are A-B, B-C, (and C-D, which is equal to B-A\ and any linear
combination of these is a period: we shall take A - B, and B-D. The latter, B-D,
is equal to

n r-i

2 / wdz -2 I wdz,

Jo Jo

which, being denoted by 4/f, gives

4J5T=4 /

JO{(1-22)(1_£222)}4

as one period. The former, A-B, is equal to

2 I wdz -2 I wdz,
Jo Jo

i

/ wdz;

/k

1|(1-

which is 2

this, being denoted by 2iK', gives

dz

dz'

where £'2 + £2=l and the relation between the variables of the integrals is

i

Hence the periods of the integral are 4K and ZiK'. Moreover, A is 2 I" wdz, which i

i J»

2 / wdz + 2 I wdz =
Jo J i

Hence the general value of f* {(I - z*) (I -

* Jordan, Cours d' 'Analyse, t. ii, p. 356.

13—2

196 PERIODICITY [104.

or

that is, 2K-u + 4mK+2niK',

where u is the integral taken from 0 to z along an assigned path, often taken to be
a straight line ; so that there are two systems of values for the integral, each containing
a doubly -infinite number of terms.

If z be denoted by $ (u) — evidently, from the integral definition, an odd function
of u — , then

so that z is a doubly-periodic function of u, the periods being 4A and 2iK'.

Now consider the function ^ = (1 -zrf. A 2-path round T does not affect ^ by way of
change, provided the curve does not include the point 1 ; hence, if zt = x (u), we have

But a z- path round the point 1 does change % into —z1; so that

X («)--* («+**}

Hence x (u\ which is an even function, has two periods, viz., 4AT and 2A' + 2i'A", whence

x(u) = x(u + 4mK+ 2nK+ 2niK').
Similarly, taking z2 = (l -Fs2)* = -f (u), it is easy to see that

so that ^ (u), which is an even function, has two periods, viz., 2 A' and 4iK' ; whence

= u

The functions <£ (u), x (u\ ^ (M) are of course sn w> cn '""> dn M respectively.
Ex. 6. To find the general value of the integral

The function to be integrated has e^ e2, e3, and co for its branch-points; and for
paths round each of them the two branches interchange.

A circuit G which restores the initial branch of the function to be integrated can
be resolved into : —

(i) Sets of double circuits round each of the branch-points alone : as before, the

value of the integral for each of these double circuits is zero.
(ii) Sets of circuits, each enclosing two of the branch-points : it is convenient to
retain circuits including oo and en oo and e.2, oc and e3, the other three
combinations being reducible to these.
The values of the integral for these three retained are respectively

E! = 2 f (4 (z - ej (z - e2} (z - e^dz = 2«1 ,
J «i

Ez=2 I {4(2-e1)(2-e2)(s-e3)}~ick=2a>2,
J 62

3 J ea

* The choice of o> for the upper limit is made on a ground which will subsequently be
considered, viz., that, when the integral is zero, z is infinite.

104.] OF ELLIPTIC INTEGRALS 197

and therefore the value of the integral for the circuit O is of the form

But E^ K2, E3 are not linearly independent. The integral of the function round any

curve in the finite part of the plane, which does not

include el5 e<2 or e3 within its boundary, is zero, by Ex. 2;

and this curve can be deformed to the shape in the figure,

until it becomes infinitely large, without changing the

value of the integral.

Since the limit of zw for \z\ = 00 is zero, the value of
the integral from oo ' to oo is zero, by § 24, II. ; and if the
description begin with a branch w, the branch at oo is -w.
The rest of the integral consists of the sum of the values Fig. 29.

round the loops, which is

because a path round a loop changes the branch of w and the last branch after describing the
loop round e3 is +w at GO', the proper value (§ 90, in). Hence, as the whole integral
is zero, we have

or say E2 =

Thus the value of the integral for any circuit Q, which restores the initial branch of w, can
be expressed in any of the equivalent forms mE^ n E3, m'E^n'E^ m"E2 + ri'Ez, where
the m's and ris, are integers.

Now any path from co to z can be resolved into a circuit fl, which restores at oo the
initial branch of w, combined with either

(i) a straight path from oo to 2,

or (ii) a loop between oo and e1} together with a straight path from oo to z.
' (The apparently distinct alternatives, of a loop between oo and e2 together with a straight
: path from oo to z and of a similar path round ea, are inclusible in the second alternative
above ; the reasons are similar to those in Ex. 5.)

fx

If u denote j ^ {^(z-ej (z-e2) (z-e3}}~*dz when the integral is taken in a straight

, line, then the value of the integral for part (i) of a path is u; and the value of the
1 integral for part (ii) of a path is El - u, the initial branch in each case for these parts being
. the initial branch of w for the whole path. Hence the most general value of the integral
for any path is

+ 2no>3 + u,

or

the two being evidently included in the form

2mo>1 + 2n(,)3±u.
If, then, we denote by z = ft>(u) the relation which is inverse to

we

In the same way as in the preceding example, it follows that

where ^ («) is - {4 (z - e^ (z - e2) (z - e3)}*.

198 SIMPLE PERIODICITY [104

The foregoing simple examples are sufficient illustrations of the multi
plicity of value of an integral of a uniform function or of a multiform
function, when branch-points or discontinuities occur in the part of the plane
in which the path of integration lies. They also shew one of the modes in
which singly-periodic and doubly-periodic functions arise, the periodicity
consisting in the addition of arithmetical multiples of constant quantities
to the argument. And it is to be noted that, as only a single value of z
is used in the integration, so only a single value of z occurs in the
inversion ; that is, the functions just obtained are uniform functions of their
variables. To the properties of such periodic functions we shall return in the
succeeding chapters.

105. We proceed to the theory of uniform periodic functions, some
special examples of which have just been considered ; and limitation will
be made here to periodicity of the linear additive type, which is only a very
special form of periodicity.

A function f(z) is said to be periodic when there is a quantity &> such
that the equation

/(* + »)=/(*)

is an identity for all values of z. Then/0 + nw) =f(z), where n is any
integer positive or negative; and it is assumed that &> is the smallest
quantity for which the equation holds, that is, that no submultiple of &> will
satisfy the equation. The quantity u> is called a period of the function.

A function is said to be simply-periodic when there is only a single
period : to be doubly-periodic when there are two periods ; and so on, the
periodicity being for the present limited to additive modification of the
argument.

It is convenient to have a graphical representation of the periodicity of a
function.

(i) For simply-periodic functions, we
take a series of points 0, A1} A2,...,
A-i, ^4_2,... representing 0, w, 2o>, ... ,
— <», — 2&>, . . . ; and through these points
we draw a series of parallel lines, dividing
the plane into bands. Let P be any
point z in the band between the lines
through 0 and through A^\ through P
draw a line parallel to OAl and measure

each equal to OA^ then all the points /

P1} P2, ... , P_i, P-2, ... are represented

by z + nco for positive and negative integral values of n. But/ (2 + &»)=/(•*)]

and therefore the value of the function at a point Pn in any of the bands is

105.]

DOUBLE PERIODICITY

199

the same as the value at P. Moreover to a point in any of the bands there
corresponds a point in any other of the bands ; and therefore, owing to the
periodic resumption of the value at the points corresponding to each point P,
it is sufficient to consider the variation of the function for points within one
band, say the band between the lines through 0 and through AI. A point P
within the band is sometimes called irreducible, the corresponding points P
in the other bands reducible.

If it were convenient, the boundary lines of the bands could be taken
through points other than Al} A2, ... ; for example, through points (m + |) &>
for positive and negative integral values of ra. Moreover, they need not be
straight lines. The essential feature of the graphic representation is the
division of the plane into bands.

(ii) For doubly-periodic functions a similar method is adopted. Let &>
and co' be the two periods of such a
function /(#), so that

/<«.+»)»/(*)-/(•+ <0;

then f(z + nw + n'w) =f(z),
where n and n' are any integers positive
or negative.

For graphic purposes, we take points
0, A-L, A2, ..., A^i, A_2, ... representing
0, ft), 2&), . . . , — to, — 2(w, . . . ; and we take
another series 0, B1} B2, . . . , B_1} B_2, . . .
representing 0, &)', 2&/, . . . , — ft/, — 2ft/, . . . ;
through the points A we draw lines
parallel to the line of points B, and
through the points B we draw lines
parallel to the line of points A. The intersection of the lines through An
and Bn> is evidently the point n&> + w'&>', that is, the angular points of the
parallelograms into which the plane is divided represent the points nco + n'w
for the values of n and n'.

Let P be any point z in the parallelogram OAfi-JS^ ; on lines through P,
parallel to the sides of the parallelogram, take points Q1} Q2, ... , Q_lt Q_2, ...
such that PQl = QiQ2= ... = ft) and points Rlt R2, ... , R_lt R_2, ... such that
PRl = R^ — . . . — to' ; and through these new points draw lines parallel to
the sides of the parallelogram. Then the variables of the points in which
these lines intersect are all represented by z + mw + mV for positive and nega
tive integral values of m and m' ; and the point represented by z + m^ + m'a)'
is situated in the parallelogram, the angular points of which are mw 4 mot',
(m + 1) &) + mw, mco + (mf + 1) ft)', and (m -f 1) &) + (m + 1) ft/, exactly as P
is situated in OA^C^. But

/ (z + m^ + Wj V) = / (z\

Fig. 31.

200 RATIO OF THE PERIODS [105.

and therefore the value of the function at such a point is the same as the
value at P. Since the parallelograms are all equal and similarly situated.
to any point in any of them there corresponds a point in OA^G^B^; and the
value of the function at the two points is the same. Hence it is sufficient to
consider the variation of the function for points within one parallelogram, say,
that which has 0, &>, o) + «', &>' for its angular points. A point P within
this parallelogram is sometimes called irreducible, the corresponding points
within the other parallelograms reducible to P ; the whole aggregate of the
points thus reducible to any one are called homologous points. And the
parallelogram to which the reduction is made is called the parallelogram of
periods.

As in the case of simply-periodic functions, it may prove convenient to
choose the position of the fundamental parallelogram so that the origin is
not on its boundary ; thus it might be the parallelogram the middle points of
whose sides are + £&>, + ^co'.

106. In the preceding representation it has been assumed that the line
of points A is different in direction from the line of points B. If &> = u + iv
and to' = u'+iv', this assumption implies that v'/u' is unequal to v/u, and
therefore that the real part of a>'/ia> does not vanish. The justification of
this assumption is established by the proposition, due to Jacobi * : —

The ratio of the periods of a uniform doubly -periodic function cannot be
real.

Let/ (2) be a function, having CD and CD' as its periods. If the ratio w'/to
be real, it must be either commensurable or incommensurable.

If it be commensurable, let it be equal to n'/n, where n and n' are
integers, neither of which is unity owing to the definition of the periods CD
and 6Dj.

Let n'/n be developed as a continued fraction, and let m'fm be the last
convergent before n'jn, where m and mf are integers. Then

n' m _ 1

n m mn'
that is, mn' - m'n = 1,

, 1 / . U> , . .. CD

so that mco ~ mco = -(mn~ run ) = - .

n x n

Therefore f(z) =f(z + m'co ~ mco'),

since m and m' are integers ; so that

-, , ~( co\

/(*)-/(' -i- s).

contravening the definition of CD as a period, viz., that no submultiple of co is a
period. Hence the ratio of the periods is not a commensurable real quantity.

* Ges. Werke, t. ii, pp. 25, 26.

106.] OF A UNIFORM DOUBLY-PERIODIC FUNCTION 201

If it be incommensurable, we express oj'/aj as a continued fraction. Let
p/q and p'/q' be two consecutive convergents : their values are separated by
the value of &>'/&>, so that we may write

v~q+ \q'~q)'

where 1 > h > 0.

Now pq <- p'q — 1, so that

- = P + —
o> q qq

where e is real and |e < 1 ; hence

, e

qa) —pa) = —, &>.

Therefore f(z) =f(z + qw — pa),

since p and q are integers ; so that

Now since &>'/&> is incommensurable, the continued fraction is unending. We
therefore take an advanced convergent, so that q' is very large. Then €- &> is

a very small quantity and z + - &> is a point infinitesimally near to z, that

is, the function / (V), under the present hypothesis, resumes its value at a
point infinitesimally near to z. Passing along the line joining these two
points infinitesimally near another, we should have / (z) constant along a
line and therefore (§ 37) constant everywhere ; it would thus cease to be a
varying function.

The ratio of the periods is thus not an incommensurable real quantity.

We therefore infer Jacobi's theorem that the ratio of the periods cannot
be real. In general, the ratio is a complex quantity ; it may, however, be a
pure imaginary*.

COROLLARY. If a uniform function have two periods wl and &>2 such that
a relation

mlwl + ra2G>2 = 0

exists for integral values of m1 and ?n2, the function is only simply-periodic.
And such a relation cannot exist between two periods of a simply-periodic
function, if m^ and ra2 be real and incommensurable ; for then the function
would be constant.

* It was proved, in Ex. 5 and Ex. 6 of § 104, that certain uniform functions are doubly-periodic.
A direct proof, that the ratio of the distinct periods of the functions there obtained is not a real
quantity, is given by Falk, Acta Math., t. vii, (1885), pp. 197—200, and by Pringsheim, Math.
Ann., t. xxvii, (1886), pp. 151—157.

202 UNIFORM [106.

Similarly, if a uniform function have three periods &>1; a>.2> o>3, connected

by two relations

.. = 0,

n1o)1 + n2a)2 + n3a)3 = 0,

where the coefficients m and n are integers, then the function is only simply-
periodic.

107. The two following propositions, also due to Jacobi*, are important
in the theory of uniform periodic functions of a single variable : —

If a uniform function have three periods w^, «2, MS such that a relation

m^i + m.2&>2 + m3w3 = 0

is satisfied for integral values ofmlt w2, m3, then the function is only a doubly-
periodic function.

What has to be proved, in order to establish this proposition, is that two
periods exist of which wl, &>2, &>3 are integral multiple combinations.

Evidently we may assume that m^, ra2, m3 have no common factor: let /
be the common factor (if any) of m.2 and m3, which is prime to m^. Then
since

and the right-hand side is an integral combination of periods, it follows that
riod.

is a fraction in its lowest terms. Change it into a continued

-~ &>! is a period.

fraction and let ^ be the last convergent before the proper value ; then
2

1

so that <l~f~P=±^f-

But o>! is a period and ^ft)! is a period; therefore q —^ Wj — pwi is a period,

or &>!// is a period, = to/ say.

Let ra2//= m2', m3/f= m/, so that m1&V + m2'&>2 + ??i3'&)3 = 0. Change

fy»

m.2'/m3 into a continued fraction, taking - to be the last convergent before the

proper value, so that

m/ r _ 1

/ i

s sms

* Ges. Werke, t. ii, pp. 27—32.

107.] DOUBLY-PERIODIC FUNCTIONS 203

Then r&>2 + sco., being an integral combination of periods, is a period. But
± &>2 = &)2 (sm2r — rm3)

= — ra>.2m3 — s (m^ + w3'&>3)

= — m^sw-i' - ma' (r&>2 + su>3) ;
also + ft)3 = &)3 (sm/ — rm3)

— sm2'o)3 + r (mjO)/

and o>! =/&)/.

Hence two periods &>/ and r<u2 + s&>3 exist of which co1} co2, &>3 are integral
multiple combinations ; and therefore all the periods are equivalent to &>/ and
r&>2 + so)3, that is, the function is only doubly-periodic.

COROLLARY. If a function have four periods <ul3 &>2, cos, &>4 connected by
two relations

m1o)1 + m2o)2 + wi3ft)3 + ra4&>4 = 0,

72J60J + W20)2 + W3ft)3 + W4«04 = 0,

where the coefficients m and w are integers, the function is only doubly-
periodic.

108. If a uniform function of one variable have three periods a)l, w.,, &>3,
then a relation of the form

m1o)l + w?2to2 + in3(i)3 = 0
must be satisfied for some integral values ofml} m2, ms.

Let a)r = ar + i@r, for r = 1, 2, 3 ; in consequence of § 106, we shall assume
that no one of the ratios of twj, <w2, w3 in pairs is real, for, otherwise, either
the three periods reduce to two immediately, or the function is a constant.
Then, determining two quantities A, and fj, by the equations

so that X and //, are real quantities and neither zero nor infinity, we have

for real values of X and p.

Then, first, if either X or fj. be commensurable, the other is also commen
surable. Let X = a/6, where a and b are integers ; then

= bo)3 — aa)},

so that fyu,&>2 is a period. Now, if b/j, be not commensurable, change it into a
continued fraction, and let p/q, p'/q be two consecutive convcrgents, so that,
as in § 106,

/ P , x
bfji,=^+ —,,

q qq

204 TRIPLY-PERIODIC UNIFORM [108.

where 1 > x > — 1. Then - &>., + -— ? is a period, and so is <w2 ; hence

q qq

'P~ ^x

IT

is a period, that is, - <a2 is a period. We may take q indefinitely large, and

then the function has an infinitesimal quantity for a period, that is, it would
be a constant under the hypothesis. Hence &/* (and therefore /*) cannot be
incommensurable, if X be commensurable; and thus X and //. are simul
taneously commensurable or simultaneously incommensurable.

CL G

If X and fj, be simultaneously commensurable, let X = j- , p = -^ , so that

a c

&)3 = r &>! + -jG>2.

o a

and therefore 6rfto3 = ac^ + bca)2,

a relation of the kind required.

If X and //. be simultaneously incommensurable, express A, as a continued
fraction ; then by taking any convergent r/s, we have

r _ x
*=*'

/Yt

where 1 > x > — 1, so that s\ — r=-:

s

by taking the convergent sufficiently advanced the right-hand side can be
made infinitesimal.

Let i\ be the nearest integer to the value of s/j,, so that, if

we have A numerically less than ^. Then

x

sat-, — ra>1 — r1w2 = — a)1 + Aw.,,
s

fp

and the quantity - Wj can be made so small as to be negligible. Hence

S

integers r, rlt s can be chosen so as to give a new period &>/(= A&>2), such
that | &)/ < \ &)2 .

We now take wl, &>2', &>3: they will be connected by a relation of the form

0>3 =X'(W1 +yLt/G)2/,

and X' and // must be incommensurable : for otherwise the substitution for
to/ of its value just obtained would lead to a relation among a>l) &>o, &>3 that
would imply commensurability of X and of p.

Proceeding just as before, we may similarly obtain a new period &>2" such
that <o2" < \ ! mz I and so on in succession. Hence we shall obtain, after n

108.] FUNCTIONS DO NOT EXIST 205

such processes, a period co2(w) such that |&)2(n)| < ^ a>*\, so that by making n

z

sufficiently large we shall ultimately obtain a period less than any assigned
quantity. Let such period be to ; then

/(*+«)-/(*),

and so for points along the co-line we have an infinite number close together
at which the function is unaltered in value. The function, being uniform,
must in that case be constant.

It thus appears that, if A. and /j, be simultaneously incommensurable, the
function is a constant. Hence the only tenable result is that A. and //. are
simultaneously commensurable, and then there is a period-equation of the

form

m^w^ + m.2o)2 + m3o)s = 0,

where m1, w2, m3 are integers.

The foregoing proof is substantially due to Jacobi (I.e.). The result can
be obtained from geometrical considerations by shewing that the infinite
number of points, at which the function resumes its value, along a line
through z parallel to the two-line will, unless the condition be satisfied, reduce
to an infinite number of points in the a)1, &)2 parallelogram which will form
either a continuous line or a continuous area, in either of which cases the
function would be a constant. But, if the condition be satisfied, then the
points along the line through z reduce to only a finite number of points.

COROLLARY I. Uniform functions of a single variable cannot have three
independent periods ; in other words, triply -periodic uniform functions of a
single variable do not exist* ; and, a fortiori, uniform functions of a single
variable with a number of independent periods greater than two do not exist.

But functions involving more than one variable can have more than two
periods, e.g., Abelian transcendents ; and a function of one variable, having
more than two periods, is not uniform.

COROLLARY II. All the periods of a uniform periodic function of a single
variable reduce either to integral multiples of one period or to linear combina
tions of integral multiples of two periods whose ratio is not a real quantity.

109. It is desirable to have the parallelogram, in which a doubly-
periodic function is considered, as small as possible. If in the parallelogram
(supposed, for convenience, to have the origin for an angular point) there be
a point a)" such that

/(* + »")=/(*)
for all values of z, then the parallelogram can be replaced by another.

* This theorem is also due to Jacobi, (I.e., p. 202, note).

206 FUNDAMENTAL PARALLELOGRAM [109.

It is evident that co" is a period of the function ; hence (§ 108) we must

have

co" = Aco + /AW' ;

and both X and /JL, which are commensurable quantities, are less than unity
since the point is within the parallelogram. Moreover, co -f co' — <»", which
is equal to (1 — A,) co + (1 — /"•)&>', is another point within the parallelogram;
and

/(* + » + «'-«")«/(*),

since co, co', co" are periods. Thus there cannot be a single such point, unless

X = \ = p.

But the number of such points within the parallelogram must be finite ;
if there were an infinite number, they would form a continuous line or a
continuous area where the uniform function had an unvarying value, and
consequently (§ 37) the function would have a constant value everywhere.

To construct a new parallelogram when all the points are known, we first
choose the series of points parallel to the co-line through the origin 0, and of
that series we choose the point nearest 0, say Al. We similarly choose the
point, nearest the origin, of the series of points parallel to the co-line and
nearest to it after the series that includes Al} say Bl : we take OA1} OB1 as
adjacent sides of the parallelogram, and these lines as the vectorial repre
sentations of the periods. No point lies within this parallelogram where the
function has the same value as at 0 ; hence the angular points of the original
parallelograms coincide with angular points of the new parallelograms.

When a parallelogram has thus been obtained, containing no internal
point fl such that the function can satisfy the equation

for all values of z, it is called a fundamental, or a primitive, parallelogram, :
and the parallelogram of reference in subsequent investigations will be
assumed to be of a fundamental character.

But a fundamental parallelogram is not unique.

Let co and co' be the periods for a given fundamental parallelogram, so
that every other period co" is of the form Aco + //-co', where A, and /* are
integers. Take any four integers a, b, c, d such that ad — lc=±l, as may
be done in an infinite variety of ways ; and adopt two new periods coj and co2,
such that

&>! = aco + bo)', co2 = ceo + d(o'.

Then the parallelogram with coj and co2 for adjacent sides is fundamental.
For we have

+ eo = do)1 — ba>2, + co' = — ccox + aco2,

and therefore any period co"
= A.CO + /uco'
= (\d - fie) wl + (— \b + fj.a) eo2, save as to signs of A, and /z.

109.] OF PERIODS 207

The coefficients of o^ and &)2 are integers, that is, the point <w" lies outside
the new parallelogram of reference; there is therefore no point in it such that

/(* + *>")=/(*),
and hence the parallelogram is fundamental.

COROLLARY. The aggregate of the angular points in one division of the
plane into fundamental parallelograms coincides with their aggregate in
any other division into fundamental parallelograms ; and all fundamental
parallelograms for a given function are of the same area.

The method suggested above for the construction of a fundamental parallelogram is
geometrical, and it assumes a knowledge of all the points w" within a given parallelogram
for which the equation/ (z -f «")=/ (z) is satisfied.

Such a point o>3 within the o^, o>2 parallelogram is given by

nil m2

<Bo= - (Bi -\ -- 0>9,

' m3 J m3 2

where »&1} m2, m3 are integers. We may assume that no two of these three integers
have a common factor; were it otherwise, say for m^ and wi2, then, as in § 107, a
submultiple of o>3 would be a period — a result which may be considered as excluded.
Evidently all the points in the parallelogram are the reduced points homologous with
w3, 2o>3, ...... , (m3 — 1)«3; when these are obtained, the geometrical construction is

possible.

The following is a simple and practicable analytical method for the construction.

Change w^/rag and mz/m3 into continued fractions; and let p/q and r/s be the
last convergents before the respective proper values, so that

mx p e m2 r f'

m3 q gm3' m3 s sm3'
where e and e' are each of them +1. Let

m"> n , M ml j , ^

q — =d + — , s^ = $+ — ,

m3 m3 m3 m3

where X and p, are taken to be less than m3, but they do not vanish because q and s are
less than m3. Then

2'eo3-^w1-(9o)2 = — (/*a>2 + f»i), *a>3-ro>2 -<£<•>! = — (Xa^ + e'tOjj) ;

U vn II io

the left-hand sides are periods, say Qx and O2 respectively, and since /u + e is not >m3 and
X + e' is not >m3, the points Q.l and Q2 determine a parallelogram smaller than the initial
parallelogram.

Thus
are equations defining new periods Qly Q2. Moreover

, . X m-. p 65 a m9 r t'o

4>-\ -- = s-^=s*-+ -, 0 + -f^ = n —? = «- + -L :

m3 m3 q qms m3 2 ms * s sm3

so that, multiplying the right-hand sides together and likewise the left-hand sides, we
at once see that X/i-ee' is divisible by ms if it be not zero: let

X/i — ee' = wi3A.

Then, as X and p are less than m3, they are greater than A; and they are prime to it,
because ee' is +1.

208 MULTIPLE [109.

Hence we have Aa>j = ^Q2- t'Ql, Aa>2 = XQ1- eiV

Since X and /u are both greater than A, let

X = X1A + X', /x = /i1A + //,
where X' and /x' are <A. Then X'/*'— «' ig divisible by A if it be not zero, say

X'p - ee = AA' ;
then X' and p.' are >A' and are prime to it. And now

A (wj — /^iO2) = /x'Q2 ~ e'^i > A (W2 "~ ^1^1) = ^-'QI ~ f®2 i

and therefore, if (a1 — /^G^Qg, <B2-X1Q1 = Q4, which are periods, we have

With Q3 and Q4 we can construct a parallelogram smaller than that constructed
with Qj and Q2.

We now have A'Q1 = fG3+//G4, A'Q.j=X'Q3 + e'fl4,

that is, equations of the same form as before. We proceed thus in successive stages :
each quantity A thus obtained is distinctly less than the preceding A, and so finally we
shall reach a stage when the succeeding A would be unity, that is, the solution of the pair
of equations then leads to periods that determine a fundamental parallelogram. It
is not difficult to prove that a>lt o>2, o>3 are combinations of integral multiples of these
periods.

If one of the quantities, such as X'/x'-ee', be zero, then X'=/x' = l, e = e'= ±1 ; and then
Q3 and O4 are identical. If e = e' = + 1, then AQ3 = Q2 - Qj , and the fundamental parallelo
gram is determined by

<V = QI + - (Q2 - %), G4' = Q2 - 1 (Q2 - Qt)-

If f = f = -1, then AQ3 = Q2+O15 so that, as A is not unity in this case, the fundamental
parallelogram is determined by Q2 and Q3.

Ex. If a function be periodic in a>1? a>2, and also in <o3 where

29co = 1 7

periods for a fundamental parallelogram are

QI = Scoj + 3o)2 - 8w3 , Q2' = 3 eoj + 2co2 - 5w3 ,
and the values of a>1} <»2, w3 in terms of O/ and Q2' are

G)2 = 2-1, a>3 = 2 Q.

Further discussion relating to the transformation of periods and of fundamental
parallelograms will be found in Briot and Bouquet's The'orie des fonctions elliptujues,
pp. 234, 235, 268—272.

110. It has been proved that uniform periodic functions of a single
variable cannot have more than two periods, independent in the sense that
their ratio is not a real quantity. If then a function exist, which has two
periods with a real incommensurable ratio or has more than two independent
periods, either it is not uniform or it is a function (whether uniform or multi
form) of more variables than one.

When restriction is made to uniform functions, the only alternative is
that the function should depend on more than one variable.

110.] PERIODICITY 209

In the case when three periods o)l, &>2, &>3 (each of the form a-f t/3) were
assigned, it was proved that the necessary condition for the existence of a
uniform function of a single variable is that finite integers mly m2, m:i can
be found such that

ra2cr2 + m3a3 = 0,

- w3/33 = 0 ;

and that, if these conditions be not satisfied, then finite integers m1} m.2, ms
can be found such that both Sma and 2m/8 become infinitesimally small.

This theorem is purely algebraical, and is only a special case of a more
general theorem as follows :

Let an, or12,..., alt r+1; a21, aw,..., a2>r+1;...; arl, «.«,..., ar,r+i be r sets of
real quantities such that a relation of the form

wia*i + ^2^2 + . • • + nr+l <xsr+i = 0

is not satisfied among any one set. Then finite integers m^,..., mr+1 can be
determined such that each of the sums

j (for 5 = 1, 2,...,r) is an infinitesimally small quantity. And, a fortiori, if
fewer than r sets, each containing r+1 quantities be given, the r+1
integers can be determined so as to lead to the result enunciated ; all that

i is necessary for the purpose being an arbitrary assignment of sets of real

| quantities necessary to make the number of sets equal to r. But the result

! is not true if more than r sets be given.

We shall not give a proof of this general theorem* ; it would follow the
lines of the proof in the limited case, as given in § 108. But the theorem
will be used to indicate how the value of an integral with more than
two periods is affected by the periodicity.

Let / be the value of the integral taken along some assigned path from
an initial point ZQ to a final point z\ and let the periods be &)1} &>2,..., &>r,
(where r > 2), so that the general value is

/ + fftjcoj + m2a)., + . . . + mrwr,

where mlt m2,..., mr are integers. Now if cos = as + i/3s, for s=l, 2,..., r,
when it is divided into its real and its imaginary parts, then finite integers
Wi, n2,..., nr can be determined such that

and n-if.

are both infinitesimal ; and then

2 ns

is infinitesimal. But the addition

of S nscos still gives a value of the integral ; hence the value can be modified

* A proof will be found in Clebsch and Gordan's Theorie der Abel'schen Functioncn, § 38.
F- 14

210 MULTIPLE PERIODICITY [110.

by infinitesimal quantities, and the modification can be repeated indefinitely.
The modifications of the value correspond to modifications of the path from
ZQ to z ; and hence the integral, regarded as depending on a single variable,
can be made, by modifications of the path of the variable, to assume any
value. The integral, in fact, has not a definite value dependent solely
upon the final value of the variable; to make the value definite, the path
by which the variable passes from the lower to the upper limit must be
specified.

It will subsequently (§ 239) be shewn how this limitation is avoided by
making the integral, regarded as a function, depend upon a proper number
of independent variables — the number being greater than unity.

Ex. 1. If F0 be the value of i — -, , (n integral), taken along an assigned path,

Jo (\-znY
and if

P = 2 I1—- ^-j(# real),

then the general value of the integral is

I \ ^ I

n

where q is any integer and mp any positive or negative integer such that 2 mp = 0.

P=I

(Math. Trip. Part II, 1889.)

Ex. 2. Prove that v= I udz, where
J o

is an algebraical function satisfying the equation

and obtain the conditions necessary and sufficient to ensure that

i) = fadz
should be an algebraical function, when u is an algebraical function satisfying an equation

(Liouville, Briot and Bouquet.)

CHAPTER X.

SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS.

111. ONLY a few of the properties of simply-periodic functions will be
given, partly because some of them are connected with Fourier's series the
detailed discussion of which lies beyond our limits, and partly because, as
will shortly be explained, many of them can at once be changed into
properties of uniform non-periodic functions which have already been
considered.

When we use the graphical method of § 105, it is evident that we need
consider the variation of the function within only a single band. Within
that band any function must have at least one infinity, for, if it had not, it
would not have an infinity anywhere in the plane and so would be a constant ;
and it must have at least one zero, for, if it had not, its reciprocal, also a
simply-periodic function, would not have an infinity in the band. The
infinities may, of course, be accidental or essential : their character is repro
duced at the homologous points in all the bands.

For purposes of analytical representation, it is convenient to use a
relation

Ziri

so that, if the point Z in its plane have R and (*)
for polar coordinates,

, „

Z = =— ; log R +

Z7TI

ft).

If we take any point A in the ^-plane and a
corresponding point a in the z-plane, then, as Z
describes a complete circle through A with the
origin as centre, z moves along a line aal} where
di is a + a). A second description of the circle
makes z move from ax to aa, where a2 = ax + &> •

Fig. 32.

and so on in succession.
14—2

212 SIMPLE PERIODICITY [111.

For various descriptions, positive and negative, the point a describes a line,
the inclination of which to the axis of real quantities is the argument of &>.

Instead of making Z describe a circle through A, let us make it describe
a part of the straight line from the origin through A, say from A, where
OA = R, to C, where 00 = R'. Then z describes a line through a perpend
icular to aal} and it moves to c where

Similarly, if any point A' on the former circumference move radially to a
point C at a distance R from the ^-origin, the corresponding z point a'
moves through a distance a'c', parallel and equal to ac : and all the points c'
lie on a line parallel to aa^. Repeated description of a ^-circumference with
the origin as centre makes z describe the whole line cCjCo.

If then a function be simply-periodic in &>, we may conveniently take
any point a, and another point a^ = a + w, through a and a^ draw straight
lines perpendicular to aa1} and then consider the function within this band.
The aggregate of points within this band is obtained by taking

(i) all points along a straight line, perpendicular to a boundary of

the band, as aa^ ;
(ii) the points along all straight lines, which are drawn through the

points of (i) parallel to a boundary of the band.

In (i), the value of z varies from 0 to co in an expression a + z, that is, in
the ^-plane for a given value of R, the angle © varies from 0 to 2?r.

In (ii), the value of log R varies from — oo to +00 in an expression
fi\

. log R + =— w, that is, the radius R must vary from 0 to oo .

2?r

Hence the band in the 0-plane and the whole of the ^-plane are made
equivalent to one another by the transformation

Now let z0 be any special point in the finite part of the band for a given
simply-periodic function, and let Z0 be the corresponding point in the Z-planej
Then for points z in the immediate vicinity of z0 and for points Z which
are consequently in the immediate vicinity of Z0, we have

Ziri

e
to

where | X differs from unity only by an infinitesimal quantity.

111.] FOURIER'S THEOREM 213

If then w, a function of z, be changed into W a function of Z, the following
relations subsist : —

When a point ZQ is a zero of w, the corresponding point ZQ is a zero

of W.

When a point z0 is an accidental singularity of w, the corresponding
point Z0 is an accidental singularity of W.

When a point z0 is an essential singularity of w, the corresponding
point Z0 is an essential singularity of W.

When a point z0 is a branch- point of any order for a function w, the
corresponding point Z0 is a branch-point of the same order for W.

And the converses of these relations also hold.

Since the character of any finite critical point for w is thus unchanged by the
transformation, it is often convenient to change the variable to Z so as to let
the variable range over the whole plane, in which case the theorems already
proved in the preceding chapters are applicable. But special account must
be taken of the point z = oo .

112. We can now apply Laurent's theorem to deduce what is practically

Fourier's series, as follows.

Let f(z) be a simply-periodic function having w as its period, and suppose
that in a portion of the z-plane bounded by any two parallel lines, the inclina
tion of which to the axis of real quantities is equal to the argument of w, the
function is uniform and has no singularities; then, at points within that
portion of the plane, the function can be expressed in the form of a converging

2n-2t

series of positive and of negative integral powers of e "° .

In figure 32, let aa^a^... and cc^... be the two lines which bound the
portion of the plane : the variations of the function will all take place within
that part of the portion of the plane which lies within one of the repre
sentative bands, say within the band bounded by ...ac... and . ..a^...: that is,
we may consider the function within the rectangle acc^a, where it has no
singularities and is uniform.

Now the rectangle acc^a in the 2-plane corresponds to a portion of the
Z-plane which, after the preceding explanation, is bounded by two circles

2iri 2irf

with the origin for common centre and of radii | e w " | and | e u ' ; and the
variations of the function within the rectangle are given by the variations of
a transformed function within the circular ring. The characteristics of the
one function at points in the rectangle are the same as the characteristics of
the other at points in the circular ring : and therefore, from the character
of the assigned function, the transformed function has no singularities and it

214 FOURIER'S THEOREM [112.

is uniform within the circular ring. Hence, by Laurent's Theorem (§ 28),
the transformed function is expressible in the form

a series which converges within the ring : and the value of the coefficient an

is given by

1

tvfj Zn+*

taken along any circle in the ring concentric with the boundaries.

Retransforming to the variable z, the expression for the original function
is

71 = + oo Zrnriz

f(z) = 2 ane~^~ .

71= -00

The series converges for points within the rectangle and therefore, as it
is periodic, it converges within the portion of the plane assigned. And the
value of an is

Zniriz
(?\P *»~ d?

\z) 6 az,

taken along a path which is the equivalent of any circle in the ring concentric
with the boundaries, that is, along any line a'c' perpendicular to the lines
which bound the assigned portion of the plane.

The expression of the function can evidently be changed into the form

Znvi, _

1 r

-±

Wj 7

where the integral is taken along the piece of a line, perpendicular to the
boundaries and intercepted between them.

If one of the boundaries of the portion of the plane be at infinity, (so that
the periodic function has no singularities within one part of the plane), then
the corresponding portion of the ^-plane is either the part within or the part
without a circle, centre the origin, according as the one or the other of the
boundaries is at oo . In the former case, the terms with negative indices
n are absent ; in the latter, the terms with positive indices are absent.

113. On account of the consequences of the relation subsisting between
the variables z and Z, many of the propositions relating to general uniform
functions, as well as of those relating to multiform functions, can be changed,
merely by the transformation of the variables, into propositions relating to
simply-periodic functions. One such proposition occurs in the preceding
section ; the following are a few others, the full development being unnecess
ary here, in consequence of the foregoing remark. * The band of reference
for the simply-periodic functions considered will be supposed to include the

113.] SIMPLY-PERIODIC FUNCTIONS 215

origin : and, when any point is spoken of, it is that one of the series of
homologous points in the plane, which lies in the band.

We know that, if a uniform function of Z have no essential singularity,
then it is a rational algebraical function, which is integral if z = cc be the
only accidental singularity and is meromorphic if there be accidental singu
larities in the finite part of the plane ; and every such function has as many
zeros as it has accidental singularities.

Hence a uniform simply-periodic function with z=cc as its sole essential
singularity has as many zeros as it has infinities in each band of the plane ;
the number of points at which it assumes a given value is equal to the number
of its zeros ; and, if the period be w, the function is a rational algebraical

ZTTIZ

function of e a , which is integral if all the singularities be at an infinite
distance and is meromorphic if some (or all) of them be in a finite part of
the plane. But any number of the zeros and any number of the infinities
may be absorbed in the essential singularity at z = oo .

The simplest function of Z, thus restricted to have the same number of
zeros as of infinities, is one which has a single zero and a single infinity in
the finite part of the plane ; the possession of a single zero and a single infinity
will therefore characterise the most elementary simply-periodic function.
Now, bearing in mind the relation

Zniz

Z=e<*,

the simplest £-pomt to choose for a zero is the origin, so that Z = 1 ; and then
the simplest ^-point to choose for an infinity at a finite distance is \w, (being
half the period), so that Z— — \. The expression of the function in the
Z-plane with 1 for a zero and — 1 for an accidental singularity is

Z~l

and therefore assuming as the most elementary simply-periodic function that
which in the plane has a series of zeros and a series of accidental singularities
all of the first order, the points of the one being midway between those of the
other, its expression is

A

2iriz

e" -I

Zniz

which is a constant multiple of tan — . Since e " is a rational fractional

CD

function of tan — , part of the foregoing theorem can be re-stated as follows: —

If the period of the function be o>, the function is a rational algebraical

function of tan — .

n

216 SIMPLY-PERIODIC [113.

Moreover, in the general theory of uniform functions, it was found con
venient to have a simple element for the construction of products, there
(§ 53) called a primary factor: it was of the type

^Z-u

where the function G ( -~ j could be a constant; and it had only one infinity

and one zero.

Hence for simply-periodic functions we may regard tan — as a typical

primary factor when the number of irreducible zeros and the (equal) number
of irreducible accidental singularities are finite. If these numbers should
tend to an infinite limit, then an exponential factor might have to be

associated with tan — ; and the function in that case might have essential
singularities elsewhere than at z = oo .

114. We can now prove that every uniform function, which has no
essential singularities in the finite part of the plane and is such that all its
accidental singularities and its zeros are arranged in groups equal and
finite in number at equal distances along directions parallel to a given
direction, is a simply-periodic function.

Let to be the common period of the groups of zeros and of singularities :
and let the plane be divided into bands by parallel lines, perpendicular to
any line representing w. Let a, b, ... be the zeros, a, /3, ... the singularities
in any one band.

Take a uniform function </> (z), simply-periodic in <w and having a single
zero and a single singularity in the band : we might take tan — as a value
of <f> (z). Then

is a simply-periodic function having only a single zero, viz., z = a and a single
singularity, viz., z — a. ; for as <f> {z} has only a single zero, there is only a single
point for which (f>(z) = <f) (a), and a single point for which <£ (z) — $ (a). Hence

is a simply-periodic function with all the zeros and with all the infinities of
the given function within the band. But on account of its periodicity it has
all the zeros and all the infinities of the given function over the whole plane ;
hence its quotient by the given function has no zero and no singularity over
the whole plane and therefore it is a constant ; that is, the given function,

114.] FUNCTIONS 217

save as to a constant factor, can be expressed in the foregoing form. It is
thus a simply-periodic function.

This method can evidently be used to construct simply-periodic functions, having
assigned zeros and assigned singularities. Thus if a function have a + mat as its zeros and
c+m'<o as its singularities, where m and m' have all integral values from — oo to +00,
the simplest form is obtained by taking a constant multiple of

TTZ 7T«

tan tan —

TTZ , TTC
tan tan —

Ex. Construct a function, simply-periodic in w, having zeros given by (m+^)o> and

)o> and singularities by (m + i)co and (m + §) co.
The irreducible zeros are ^co and f w ; the irreducible singularities are \u> and §«. Now

f.TTZ \ ( TTZ , \

I tan tan ATT I I tan tan |TT I

/ \ <" / \ M /

7TZ \ ( TTZ „ \

tan tan JTT ] ( tan tan |TT I

/ \ /

is evidently a function, initially satisfying the required conditions. But, as tari^r is
infinite, we divide out by it and absorb it into A' as a factor ; the function then takes
the form

1 + tan -

3-tan'7^

60

We shall not consider simply-periodic functions, which have essential
singularities elsewhere than at z = <x> ; adequate investigation will be found
in the second part of Guichard's memoir, (I.e., p. 147). But before leaving the
consideration of the present class of functions, one remark may be made. It
was proved, in our earlier investigations, that uniform functions can be
expressed as infinite series of functions of the variable and also as infinite
products of functions of the variable. This general result is true when the
functions in the series and in the products are simply-periodic in the same
period. But the function, so represented, though periodic in that common
period, may also have another period : and, in fact, many doubly-periodic
functions of different kinds (§ 136) are often conveniently expressed as infinite
converging series or infinite converging products of simply-periodic functions.

Any detailed illustration of this remark belongs to the theory of elliptic functions : one
simple example must suffice.

, ima'

Let the real part of - - be negative, and let q denote e " ; then the function

being an infinite converging series of powers of the simply-periodic function e " , is finite
everywhere in the plane. Evidently 6 (z) is periodic in o>, so that

= 6 (z).

218 DOUBLY-PERIODIC [114.

°

Again, 0(s + «»') = 2

the change in the summation so as to give $ (z) being permissible because the extreme
terms for the infinite values of n can be neglected on account of the assumption with
regard to q. There is thus a pseudo-periodicity for 6(z) in a period <•>'.

Similarly, if 0s(z)= q* e

2J7TZ

63(z + <a') = -e " 6(z).

Then 63(z) -r-d(z) is doubly-periodic in w and 2co', though constructed only from
functions simply-periodic in w : it is a function with an infinite number of irreducible
accidental singularities in a band.

115. We now pass to doubly-periodic functions of a single variable, the
periodicity being additive. The properties, characteristic of this important
class of functions, will be given in the form either of new theorems or
appropriate modifications of theorems, already established ; and the develop
ment adopted will follow, in a general manner, the theory given by Liouville*.
It will be assumed that the functions are uniform, unless multiformity be
explicitly stated, and that all the singularities in the finite part of the plane
are accidental "f*.

The geometrical representation of double-periodicity, explained in § 105,
will be used concurrently with the analysis; and the parallelogram of
periods, to which the variable argument of the function is referred, is a
fundamental parallelogram (§ 109) with periods J 2co and 2&>'. An angular
point £0 for the parallelogram of reference can be chosen so that neither a
zero nor a pole of the function lies on the perimeter; for the number
of zeros and the number of poles in any finite area must be finite,
otherwise they would form a continuous line or a continuous area, or thej
would be in the vicinity of an essential singularity. This choice will, ir

* In his lectures of 1847, edited by Borchardt and published in Crelle, t. Ixxxviii, (1880), pp.
277 — 310. They are the basis of the researches of Briot and Bouquet, the most complet
exposition of which will be found in their Theorie des fonctions elliptiques, (2nd ed.), pp.
239—280.

t For doubly-periodic functions, which have essential singularities, reference should be made
to Guichard's memoir, (the introductory remarks aud the third part), already quoted on p. 147, note.

J The factor 2 is introduced merely for the sake of convenience.

115.] FUNCTIONS 219

general, be made ; but, in particular cases, it is convenient to have the origin
as an angular point of the parallelogram and then it not infrequently occurs
that a zero or a pole lies on a side or at a corner. If such a point lie on a side,
the homologous point on the opposite side is assigned to the parallelogram
which has that opposite side as homologous; and if it be at an angular point,
the remaining angular points are assigned to the parallelograms which have
them as homologous corners.

The parallelogram of reference will therefore, in general, have z0, z0 + 2&>,
z0 + 2&/, z0 + 2&> + 2&>' for its angular points ; but occasionally it is desirable
to .take an equivalent parallelogram having z0 ± &> + &>' as its angular
points.

When the function is denoted by </> (2), the equations indicating the
periodicity are

<£ (z + 2<w) = (f> (z) = (f> (z + 2&/).

116. We now proceed to the fundamental propositions relating to
doubly-periodic functions.

I. Every doubly -periodic function must have zeros and infinities within
the fundamental parallelogram.

For the function, not being a constant, has zeros somewhere in the plane
and it has infinities somewhere in the plane ; and, being doubly-periodic, it
experiences within the parallelogram all the variations that it can have over
the plane.

COROLLARY. The function cannot be a rational integral function of z.

For within a parallelogram of finite dimensions an integral function has
no infinities and therefore cannot represent a doubly-periodic function.

An analytical form for <j) (z) can be obtained which will put its singu
larities in evidence. Let a be such a pole, of multiplicity n ; then we know
that, as the function is uniform, coefficients A can be determined so that the
function

f(* ~ (z-a)n~ (z-a)n-1~'"~(z-a)2~ z^a

is finite in the vicinity of a ; but the remaining poles of <j> (z) are singularities
of this modified function. Proceeding similarly with the other singularities
b, c,..., which are finite in number and each of which is finite in degree, we
have coefficients A, B, C,... determined so that

A^< V i? K'

9 (z) — 2, f Z T r

is finite in the vicinity of every pole of <f) (z) within the parallelogram and
therefore is finite everywhere within the parallelogram. Let its value be

220 PROPERTIES [116.

%(X); then for points lying within the parallelogram, the function <f>(z) is
expressed in the form

+ A* +

^ 1 1

A,

T T ;

2 — a (

ft

+ 1 i

\9 1 ' ' ' ' /

z - a> (
B2

z - a)n
Bm

X. ' /

£—6 (

7 \ 0 ' • • • 1 /

z-b)m

H,

_L _L

#2

S±i

z-h^ (z-h? ' r (z-h)1'

But though <£ (^) is periodic, ^ (2?) is not periodic. It has the property of
being finite everywhere within the parallelogram ; if it were periodic, it
would be finite everywhere, and therefore could have only a constant value ;
and then <f> (z) would be an algebraical meromorphic function, which is not
periodic. The sum of the fractions in $ (z) may be called the fractional
part of the function : owing to the meromorphic character of the function,
it cannot be evanescent.

The analytical expression can be put in the form
(z - a)~n (z - 6)-™. . .(z - h)~l F(z\

where F(z) is finite everywhere within the parallelogram. If a, /3, ..., ij be
all the zeros, of degrees v, p, ..., X, within the parallelogram, then

F(z) = (z-a)v(z-py ...(z-^G(z\

where G (z) has no zero within the parallelogram ; and so the function can
be expressed in the form

(z-a)n(z-b}m...(z-h)1 G^'

where G (z) has no zero and no infinity within the parallelogram or on its
boundary ; and G {z) is not periodic.

The order of a doubly-periodic function is the sum of the multiplicities
of all the poles which the function has within a fundamental parallelogram;
and, the sum being n, the function is said to be of the nth order. All
these singularities are, as already remarked, accidental; it is convenient
to speak of any particular singularity as simple, double, . . . according to its
multiplicity.

If two doubly-periodic functions u and v be such that an equation

is satisfied for constant values of A, B, C, the functions are said to be
equivalent to one another. Equivalent functions evidently have the same
accidental singularities in the same multiplicity.

II. The integral of a doubly-periodic function round the boundary of a
fundamental parallelogram is zero.

116.]

OF DOUBLY-PERIODIC FUNCTIONS

221

Let ABCD be a fundamental parallelogram, the boundary of it being
taken so as to pass through no pole of the
function. Let A be z0, B be z0+2ca, and* <=

D be z0 + 2a)': then any point in AB is / °

/Q* Q,

where £ is a real quantity lying between 0 and 1 ;
and therefore the integral along AB is

rl

Any point in EG is z0 + 2<w + 2&>'£, where £ is a real quantity lying between 0
and 1 ; therefore the integral along BC is

(o 'dt,

o

since <^> is periodic in 2&).

Any point in DC is s + 2o>' + 2<wZ, where < is a real quantity lying
between 0 and 1 ; therefore the integral along CD is

f°

J 1

2ft)'

= - I

J o

Similarly, the integral along DA is

= - I cf> Oo + 2o>'«) 2w'^.

J o

Hence the complete value of the integral, taken round the parallelogram, i

fi
= <j>(z0

Jo

which ^ is manifestly zero, since each of the integrals is the integral of
a continuous function.

COROLLARY. Let ty(z) be any uniform function of zt not necessarily
doubly-periodic, but without singularities on the boundary. Then the

* The figure implies that the argument of w' is greater than the argument of w, a
hypothesis which, though unimportant for the present proposition, must be taken account of
hereafter (e.g., § 129).

222 INTEGRAL RESIDUE [116.

integral jty (z) dz taken round the parallelogram of periods is easily seen
to be

n ri

•^ (z(} + Scot) 2udt + I ^(z0 + 2a> + 2m't) 2a>'dt

Jo J o

ri ri

- V (*o + 2o>' + 2a>t) 2(odt - ^ (z0 + 2to't) 2w'dt ;

Jo Jo

or, if we write

/• ri ri

then U- (2) ^ = I I/TJ (>0 + 2w't) 2m dt - ^ (z, + 2wt) 2(odt,

J Jo Jo

where on the left-hand side the integral is taken positively round the
boundary of the parallelogram and on the right-hand side the variable t
in the integrals is real.

The result may also be written in the form

r rD rx

\-^r(z)dz=\ ^ (z) dz — I -»K (z) dz,

J J A J A

the integrals on the right-hand side being taken along the straight lines AD
and AB respectively.

Evidently the foregoing main proposition is established, when -^ (£) and
T/r2 (f) vanish for all values of £.

III. If a doubly -periodic function $(z) have infinities Oj, a2, ... within
the parallelogram, and if Al, A2, ... be the coefficients of (z — e^)"1, (z — a^r1, . . .
respectively in the fractional part of (j> (z) when it is expanded in the parallelo
gram, then

A1 + A2+...=0.

As the function <f>(z) is uniform, the integral f(f>(z)dz is, by (§ 19, II.), the
sum of the integrals round a number of curves each including one and only
one of the infinities within that parallelogram.

Taking the expression for (f>(z) on p. 220, the integral Amf(z — a)~mdz
round the curve enclosing a is 0, if m be not unity, and is Z>jriAl, if m be
unity; the integral Kmf(z — k)~mdz round that curve is 0 for all values of m
and for all points k other than a ; and the integral /^ (z) dz round the curve
is zero, since % (z) is uniform and finite everywhere in the vicinity of a. Hence
the integral of <£ (z) round a curve enclosing c^ alone of all the infinities is

Similarly the integral round a curve enclosing a.2 alone is 27riA.2; and so
on, for each of the curves in succession.

Hence the value of the integral round the parallelogram is

2-rnZA.

116.] OF FUNCTIONS OF THE SECOND ORDER 223

But by the preceding proposition, the value of /(/> (2) dz round the parallelo
gram is zero ; and therefore

This result can be expressed in the form that the sum of the residues* of a
doubly -periodic function relative to a fundamental parallelogram of periods
is zero.

COROLLARY 1. A doubly-periodic function of the first order does not

exist.

Let such a function have a for its single simple infinity. Then an
expression for the function within the parallelogram is

A

^-a + *^>

where ^ (2) is everywhere finite in the parallelogram. By the above propo
sition, A vanishes ; and so the function has no infinity in the parallelogram.
It therefore has no infinity anywhere in the plane, and so is merely a
constant : that is, qua function of a variable, it does not exist.

COROLLARY 2. Doubly-periodic functions of the second order are of two
classes.

As the function is of the second order, the sum of the degrees of the
infinities is two. There may thus be either a single infinity of the second
degree or two simple infinities.

In the former case, the analytical expression of the function is

where a is the infinity of the second degree and ^ (z) is holomorphic within
the parallelogram. But, by the preceding proposition, A1 = 0; hence the
analytical expression for a doubly-periodic function with a single irreducible
infinity a of the second degree is

(z - of T * v

within the parallelogram. Such functions of the second order, which have
only a single irreducible infinity, may be called the first class.
In the latter case, the analytical expression of the function is

where c, and c2 are the two simple infinities and x(z} ig finite within the
parallelogram. Then

See p. 42.

224 PROPERTIES OF FUNCTIONS [116.

so that, if Cl = - C.2 = C, the analytical expression for a doubly-periodic
function with two simple irreducible infinities a1 and «2 ig

n
G

( 1 1

( -

\z-a-L z -

within the parallelogram. Such functions of the second order, which have
two irreducible infinities, may be called the second class.

COROLLARY 3. If within any parallelogram of periods a function is
only of the second order, the parallelogram is fundamental.

COROLLARY 4. A similar division of doubly -periodic functions of any
order into classes can be effected according to the variety in the constitution of
the order, the number of classes being the number of partitions of the order.

The simplest class of functions of the nth order is that in which the
functions have only a single irreducible infinity of the nth degree. Evi
dently the analytical expression of the function within the parallelogram is

G, G, Gn

(z - a)2 (z - a)3 (z - a)n * ^ ''

where ^ (z) is holomorphic within the parallelogram. Some of the coefficients
G may vanish ; but all may not vanish, for the function would then be finite
everywhere in the parallelogram.

It will however be seen, from the next succeeding propositions, that the
division into classes is of most importance for functions of the second
jrder.

IV. Two functions, which are doubly-periodic in the same periods*, and
which have the same zeros and the same infinities each in the same degrees
respectively, are in a constant ratio.

Let <f) and ^ be the functions, having the same periods; and let a of
degree v, /3 of degree fi, ... be all the irreducible zeros of <£ and T/T; arid a of
degree n, b of degree m, ... be all the irreducible infinities of <f> and of ty.
Then a function G (z), without zeros or infinities within the parallelogram,
exists such that

, , , = (z-a)v(z-py ... G _

and another function H(z), without zeros or infinities within the parallelo
gram, exists such that

Hence *(*)_<?(*)

-

Now the function on the right-hand side has no zeros in the parallelogram,
for G has no zeros and H has no infinities ; and it has no infinities in the

* Such functions will be called homoperiodic.

116.] OF THE SECOND ORDER 225

parallelogram, for G has no infinities and H has no zeros : hence it has
neither zeros nor infinities in the parallelogram. Since it is equal to the
function on the left-hand side, which is a doubly-periodic function, it has no
zeros and no infinities in the whole plane ; it is therefore a constant, say
A. Thus*

V. Two functions of the second order, doubly -periodic in the same periods
and having the same infinities, are equivalent to one another.

If one of the functions be of the first class in the second order, it has one
irreducible double infinity, say at a ; so that we have

where %(z) is finite everywhere within the parallelogram. Then the other
function also has z = a for its sole irreducible infinity and that infinity is of
the second degree ; therefore we have

TT

where ^ (z) is finite everywhere within the parallelogram. Hence

Now x and %x are finite everywhere within the parallelogram, and therefore
so is H% — Gfo. But H% — Gfo, being equal to the doubly-periodic function
H(j) — Gijr, is therefore doubly-periodic ; as it has no infinities within the
parallelogram, it consequently can have none over the plane and therefore it
is a constant, say 7. Thus

proving that the functions <j> and ty are equivalent.

If on the other hand one of the functions be of the second class in the
second order, it has two irreducible simple infinities, say at 6 and c, so that
we have

where 6(z) is finite everywhere within the parallelogram. Then the other
function also has z = b and z = c for its irreducible infinities, each of them
being simple ; therefore we have

where 6l (z) is finite everywhere within the parallelogram. Hence

(z) - Cty (z) = De (z) - Cei (z}.

* This proposition is the modified form of the proposition of § 52, when the generalising
exponential factor has been determined so as to admit of the periodicity.

F. 15

226 IRREDUCIBLE ZEROS [116.

The right-hand side, being finite everywhere in the parallelogram, and equal
to the left-hand side which is a doubly-periodic function, is finite everywhere
in the plane ; it is therefore a constant, say B, so that

proving that <£ and ty are equivalent to one another.

It thus appears that in considering doubly-periodic functions of the second
order, homoperiodic functions of the same class are equivalent to one another
if they have the same infinities ; so that, practically, it is by their infinities
that homoperiodic functions of the second order and the same class are dis
criminated.

COROLLARY 1. If two equivalent functions of tlie second order have one
zero the same, all their zeros are the same.

For in the one class the constant /, and in the other class the constant B,
is seen to vanish on substituting for z the common zero ; and then the two
functions always vanish together.

COROLLARY 2. If two functions, doubly-periodic in the same periods but
not necessarily of the second order, have the same infinities occurring in such a, j
way that the fractional parts of the two functions are the same except as to a
constant factor, the functions are equivalent to one another. And if, in
addition, they have one zero common, then all their zeros are common, so
that the functions are then in a constant ratio.

COROLLARY 3. If two functions of the second order, doubly-periodic in(
the same periods, have their zeros the same, and one infinity common, they are ^
in a constant ratio.

VI. Every doubly -periodic function has as many irreducible zeros as it
has irreducible infinities.

Let <£ (z) be such a function. Then

z +h — z

is a doubly-periodic function for any value of h, for the numerator is doubly-
periodic and the denominator does not involve z ; so that, in the limit when
h = 0, the function is doubly-periodic, that is, </>' (z) is doubly-periodic.

Now suppose <f>(z) has irreducible zeros of degree m1 at a1} ra2 at a2, ...,
and has irreducible infinities of degree /^ at «1} yu,2 at «2, ... ; so that the
number of irreducible zeros is Wj + ra2 + . . . , and the number of irreducible
infinities is ^1 + /i2 + ..., both of these numbers being finite. It has been
shewn that <£ {z) can be expressed in the form

116.] AND IRREDUCIBLE INFINITIES 227

whore F(z) has neither a zero nor an infinity within, or on the boundary of,
the parallelogram of reference.

Since F(z) has a value, which is finite, continuous and different from zero

Tjlt / \

everywhere within the parallelogram or on its boundary, the function -p4-r

* W
is not infinite within the same limits. Hence we have

rr - ~ — ...

9 (z) z—a± z — «2

+ -* + =*. + ..

z — ttj z — a2

where g (z) has no infinities within, or on the boundary of, the parallelogram
of reference. But, because <f> (z) and <f> (z) are doubly-periodic, their quotient
is also doubly-periodic ; and therefore, applying Prop. II., we have

m^ + w2 + . . . — ^ — p2 — . . . = 0,
that is, m1+m2 + ... = fj,! + fi2+ ...,

or the number of irreducible zeros is equal to the number of irreducible
infinities.

COROLLARY I. The number of irreducible points for which a doubly -
periodic function assumes a given value is equal to the number of irreducible
zeros.

For if the value be A, every infinity of $(z) is an infinity of the doubly-
periodic function $ (z) — A ; hence the number of the irreducible zeros of the
latter is equal to the number of its irreducible infinities, which is the same as
the number for <£ (z} and therefore the same as the number of irreducible
zeros of <£ (z). And every irreducible zero of <£ (z} — A is an irreducible
point, for which <£ (z) assumes the value A.

COROLLARY II. A doubly-periodic function with only a single zero does
not exist; a doubly -periodic function of the second order has two zeros; and,
generally, the order of a function can be measured by its number of irreducible
zeros.

Note. It may here be remarked that the doubly-periodic functions
(§ 115), that have only accidental singularities in the finite part of the
plane, have z = oo for an essential singularity. It is evident that for infinite
values of z, the finite magnitude of the parallelogram of periods is not
recognisable ; and thus for z = GO the function can have any value, shewing
that z = oo is an essential singularity.

VII. Let a1} a2)... be the irreducible zeros of a function of degrees
w1; m2, ... respectively ; a1} «2, ... its irreducible infinities of degrees /^, /u,2, ...
respectively; and z1,z2,... the irreducible points where it assumes a value c,
which is neither zero nor infinity, their degrees being M1} M.2) ... respectively.

15—2

228 IRREDUCIBLE ZEROS [116.

Then, except possibly as to additive multiples of Hie periods, the quantities
2 mrar, 2 UrCir and 2 Mrzr are equal to one another, so that

r=l r=l r=l

2 mrar = 2 Mrzr = 2 prctr (mod. 2o>, 2&/)-

r=l r=l r=l

Let (/> (/) be the function. Then the quantities which occur are the sums
of the zeros, the assigned values, and the infinities, the degree of each being
taken account of when there is multiple occurrence ; and by the last
proposition these degrees satisfy the relations

The function <f)(z) — c is doubly-periodic in 2«u and 2&>' ; its zeros are
z1} z.2, ... of degrees M1} M^,... respectively; and its infinities are ctl, «2, ... of
degrees /i1} yn2, •••, being the same as those of <£(Y). Hence there exists a
function G(z), without either a zero or an infinity lying in the parallelogram
or on its boundary, such that </> 0) - c can be expressed in the form

^l*1C.(*I*a>!'" G (*)

for all points not outside the parallelogram ; and therefore, for points in that

region

<f>'(Y) ^ Mr ^ *r G'(z)

\ /~** / \ •

/ \

<j)(z) — C r=l Z — Zr Z— O.r

Hence

z$(z) ~ Mrz v prz zG' (z)

. . >. - — 2< - --- 2* --- 1 — .~ , .
$(z) — C r=l z — zr Z— ar W (*)

= 2 Jfr+ 2

,

~r

* ~r /-v / -. ,

=\Z— Zr Z—OLr (r(z)

2 Mr= 2 nr.

r=l r=l

Integrate both sides round the boundary of the fundamental parallelogram.
Because G (z) has no zero and no infinity in the included region and does not
vanish along the curve, the integral

'zG'(z)

I

dz

G(z)

vanishes. But the points z{ and 04 are enclosed in the area ; and therefore
the value of the right-hand side is

2iri 2 Mrzr — Ziri 2 /V*r,

so

that

\Z) — c
the integral being extended round the parallelogram.

116.] AND IRREDUCIBLE INFINITIES 229

zd>' (z)

Denoting the subject of integration , by/(^), we have

<p(z) — c

-/«=*" -

and therefore, by the Corollary to Prop. II., the value of the foregoing
integral is

*• r £¥-*-** r £¥-*•

JA<f>(Z)-C JA(j)(z)-G

the integrals being taken along the straight lines AD and AB respectively
(fig. 33, p. 221).

Let w — <f)(z) — c; then, as z describes a path, w will also describe a single
path as it is a uniform function of z. When z moves from A to D, w moves
from (j>(A)-c by some path to (f>(D) — c, that is, it returns to its initial
position since <f> (D) = <f> (A) ; hence, as z describes AD, w describes a simple
closed path, the area included by which may or may not contain zeros and
infinities of w. Now

dw = <f>' (z) dz,

CD <£' (z\

and therefore the integral I ,,\ dz is equal to
* JAJ>(*)-C

I

dw
w

taken in some direction round the corresponding closed path for w. This
integral vanishes, if no w-zero or w-infinity be included within the area
bounded by the path ; it is + Im'iri, if m be the excess of the number of
included zeros over the number of included infinities, the + or — sign being
taken with a positive or a negative description ; hence we have

where m is some positive or negative integer and may be zero. Similarly

where n is some positive or negative integer and may be zero.

Thus 27Ti (2,MrZr ~ 2/V*,-) = 2w . 2w7n — 2a)' . Smri,

and therefore ^Mrzr — ^prir = 2ma> — 2?io>'

= 0 (mod. 2&), 2o>').

Finally, since ^Mrzr = 2/v*r whatever be the value of c, for the right-hand

230 DOUBLY-PERIODIC FUNCTIONS [116.

side is independent of c, we may assign to c any value we please. Let the
value zero be assigned ; then ^Mrzr becomes Smrar, so that

^mrar = "2/j,rf*r (mod. 2&>, 2&/).

The combination of these results leads to the required theorem*, expressed
by the congruences

2 mrar = 2 Mrzr = 2 ^r^r (mod. 2o>, 2&>').

r=l r=l r=l

Note. Any point within the parallelogram can be represented in the
form z0 + a2&> + 62&>', where a and 6 are real positive quantities less than

unity. Hence

2 Mrzr = Az'2a> + Bz2a>/ + z£Mr,

where J. and B are real positive quantities each less than 27lfr, that is, less
than the order of the function.

In particular, for functions of the second order, we have

z1 + z, = Az 2&> + Bz 2&/ + 2.2-0,
where Az and Bz are positive quantities each less than 2. Similarly, if a and

b be the zeros,

a + b = Aa 2w + £a 2w' + 2*o,

where J.ffl and Ba are each less than 2 ; hence, if

^i + ^2 — a — b — w2w + m'2o>',

then w may have any one of the three values - 1, 0, 1 and so may m', the
simultaneous values not being necessarily the same.

Let a and ft be the infinities of a function of the second class ; then
a + /3 — a — b = ?i2&) + n"2w',

where n and ri may each have any one of the three values — 1, 0, 1. By
changing the origin of the fundamental parallelogram, so as to obtain a
different set of irreducible points, we can secure that n and n' are zero,

and then

a + @ = a+b.

Thus, if n be 1 with an initial parallelogram, so that

a + /3 = a + &+2&>,

we should take either /3 - 2&> = {¥, or a - 2&> = a', according to the position of
a and /3, and then have a new parallelogram such that
a + @' = a + b, or a' + ft = a + b.

The case of exception is when the function is of the first class and has a
repeated zero.

* The foregoing proof is suggested by Konigsberger, Theorie der elliptischen Functionen,
t. i, p. 342 ; other proofs are given by Briot and Bouquet and by Liouville, to whom the adopted
form of the theorem is due. The theorem is substantially contained in one of Abel's general
theorems in the comparison of transcendents.

116.] OF THE SECOND ORDER 231

VIII. Let $ (z) be a doubly -periodic function of the second order. If 7
be the one double infinity when the function is of the first class, and if a and ft
be the two simple infinities when the function is of the second class, then in the
former case

and in the latter case </> (z) — <£ (a + (3 — z).

Since the function is of the second order, so that it has two irreducible
infinities, there are two (and only two) irreducible points in a fundamental
parallelogram at which the function can assume any the same value : let
them be z and z'.

Then, for the first class of functions, we have
z + z' = 27

= 27 + 2mo> + 2wa>',

where m and n are integers ; and then, since <f)(z) = <j> (z'} by definition of z
and /, we have

<£ (z) = <£ (27 - z + 2ma)

= 0(27-4
For the second class of functions, we have

z + z = a. + /3

so that, as before,

(/> (z) = </> (a + /3 - z + 2ma) + 2wa>')

117. Among the functions which have the same periodicity as a given
function </> (z), the one which is most closely related to it is its derivative
<£' (z). We proceed to find the zeros and the infinities of the derivative of a
function, in particular, of a function of the second order.

Since (f> (z) is uniform, an irreducible infinity of degree n for </> (z) is an
irreducible infinity of degree n -f 1 for §' (z). Moreover <£' (z), being uniform,
has no infinity which is not an infinity of </> (z) ; thus the order of <£' (z) is
2(?i + l) or its order is greater than that of cj>(z) by an integer which
represents the number of distinct irreducible infinities of <£ (z), no account
being taken of their degree. If, then, a function be of order m, the order of
its derivative is not less than m + 1 and is not greater than 2m.

Functions of the second order either possess one double infinity so that
within the parallelogram they take the form

—
and then <j>' (z) = — - — + %' (*),

232 ZEROS OF THE DERIVATIVE [117.

that is, the infinity of (f>(z) is the single infinity of tf>' ' (z) and it is of the
third degree, so that cf>' (z) is of the third order ; or they possess two simple
infinities, so that within the parallelogram they take the form

and then f W = - G - - _ + x' (,),

that is, each of the simple infinities of <£ (z) is an infinity for </>' (z) of the
second degree, so that <£' (z) is of the fourth order.

It is of importance (as will be seen presently) to know the zeros of
the derivative of a function of the second order.

For a function of the first class, let 7 be the irreducible infinity of the
second degree ; then we have

and therefore $'(2) = — </>' (^7 — z).

Now </>' (z) is of the third order, having 7 for its irreducible infinity in the
third degree : hence it has three irreducible zeros.

In the foregoing equation, take z = 7 : then

</>' (7) = -$' (7),

shewing that 7 is either a zero or an infinity. It is known to be the only
infinity of <£' (z).

Next, take z = 7 + &> ; then

<£' (7 + &)) = — $' (7 — a>)

= - <£' (7 + G>),

shewing that 7 + &> is either a zero or an infinity. It is known not to be an
infinity ; hence it is a zero.

Similarly 7 + &/ and 7 + <u + &/ are zeros. Thus three zeros are obtained,
distinct from one another ; and only three zeros are required ; if they be not
within the parallelogram, we take the irreducible points homologous with
them. Hence :

IX. The three zeros of the derivative of a function, doubly -periodic in
2eo and 2eo' and having 7 for its double (and only) irreducible infinity, are

7 + &), 7 + eo', 7 + w + w .

For a function of the second class, let a and /3 be the two simple
irreducible infinities; then we have

and therefore <f>' (z)= — <f>' (a + ft — z).

117.] OF A DOUBLY-PERIODIC FUNCTION 233

Now (j) (z) is of the fourth order, having a and ft as its irreducible
infinities each in the second degree ; hence it must have four irreducible
zeros.

In the foregoing equation, take z = \(VL + ft) ; then

shewing that | (a + /3) is either a zero or an infinity. It is known not to be
an infinity ; hence it is a zero.

Next, take z = £ (a + (3) + w ; then

f(}(«t£)+«} --+'{*(«+£)-••]

= - <£' & (a + £) - to + 2&>j

—.+'{*<«+£)+••},

shewing that |(a + /3) + &> is either a zero or an infinity. As before, it is
a zero.

Similarly i (a + /3) + &>' and i (a + /3) -f &> + &>' are zeros. Four zeros are
thus obtained, distinct from one another; and only four zeros are required.
Hence :

X. The four zeros of the derivative of a function, doubly-periodic in 2&>
and 2o)' and having a and /3 for its simple (and only) irreducible infinities, are

i(a + /3), i(a + /3) + a>, i(a + /3) + ft>', |- (a + /3) + w + a/.

The verification in each of these two cases of Prop. VII., that the sum of
the zeros of the doubly-periodic function <£' (z) is congruent with the sum of
its infinities, is immediate.

Lastly, it may be noted that, if zl and z^ be the two irreducible points for
which a doubly -periodic function of the second order assumes a given value,
then the values of its derivative for z1 and for z% are equal and opposite. For

(j> (z) = <f> (a + /3 - z) = cf> (z, + z.2 - z),
since zl + z., = a + (3 ; and therefore

<f> (z) = -$' (z, + z.2- z),
that is, <£' (zl) = — </>' (z2),

which proves the statement.

118. We now come to a different class of theorems.

XI. Any doubly -periodic function of the second order can be expressed
algebraically in terms of an assigned doubly-periodic function of the second
order, if the periods be the same.

The theorem will be sufficiently illustrated and the line of proof
sufficiently indicated, if we express a function (/> (z) of the second class, with
irreducible infinities a, ft and irreducible zeros a, b such that a + (3 = a + b, in

234 FUNCTIONS [118.

terms of a function <£ of the first class with 7 as its irreducible double
infinity.

n .. , ,.

Consider a function

Q (z + h) _

A zero of <X> (z + h) is neither a zero nor an infinity of this function ; nor
is an infinity of <1> (z + h) a zero or an infinity of the function. It will have
a and 6 for its irreducible zeros, if

a + h = h',
b + h + h' = 27 ;

and these will be the only zeros, for <E> is of the second order. It will have o
and yS for its irreducible infinities, if

and these will be the only infinities, for <£ is of the second order. These
equations are satisfied by

Hence the assigned function, with these values of h, has the same zeros
and the same infinities as $>(z); and it is doubly-periodic in the same periods.
The ratio of the two functions is therefore a constant, by Prop. IV., so that

c|> (z + h) — <I> (h')

If the expression be required in terms of <& (z) alone and constants, then
<j> (z 4. h} must be expressed in terms of <I> (z) and constants which are values
of <X> (z) for special values of z. This will be effected later.

The preceding proposition is a special case of a more general theorem
which will be considered later ; the following is another special case of that
theorem : viz. :

XII. A doubly -periodic function with any number of simple infinities can
be expressed either as a sum or as a product, of functions of the second order
and the second class which are doubly-periodic in the same periods.

Let «j, «2, ..., an be the irreducible infinities of the function <£, and
suppose that the fractional part of <t> (z) is

•A-i , A2 [ ^ i-+ ^n

z — ttj z — «2 z — an '

with the condition A1 + A2 + + An = Q. Let <j>n(z) be a function,

doubly-periodic in the same periods, with a,-, a,- as its only irreducible infinities,

118.] OF THE SECOND ORDER 235

supposed simple; where i and j have the values 1, ,n. Then the

fractional parts of the functions ^>j, (z), <£23 (z), . . . are

0,

G,

i z — a.,

I
\z — «2 ^ — «,

respectively; and therefore the fractional part of

^!^ / \ ,
• 0» W +

is Al An- An~l

z-a.! Z-CL, z-cin-T. z-an

•Ai An_^ An

= - -+...+- - + — ^,

Z-Cl! Z- «„_! Z - Ctn

n

since S -4* = 0. This is the same as the fractional part of <l> (z); and therefore

- <^>23 (f) - ... - -~

has no fractional part. It thus has no infinity within the parallelogram ; it
is a doubly-periodic function and therefore has no infinity anywhere in the
plane; and it is therefore merely a constant, say B. Hence, changing the
constants, we have

$>(z)-B^(z}-B.><t>v(z)-...-Bn-,<t>n-,,n(z} = B,

giving an expression for <$> (z} as a linear combination of functions of the
second order and the second class. But as the assignment of the infinities is
arbitrary, the expression is not unique.

For the expression in the form of a product, we may denote the n
irreducible zeros, supposed simple, by «!,...,«„. We determine n - 2 new
irreducible quantities c, such that

C2=

Cn—2 — &n—\ ~r Cn—3 ~ Q"n—i >
Cln = ttn + C_ — Q"—

n

this being possible because 2 o^ = 2 ar ; and we denote by $ (z ; a, ft ; e, f) a

»•=! r=l

function of .gr, which is doubly-periodic in the periods of the given function,

ALGEBRAICAL RELATIONS [118.

has a and $ for simple irreducible infinities and has e and / for simple
irreducible zeros. Then the function

<f)(z; al5 «2 ; «i, Ci) <f> (z ; as, ci ; 0-2, c2) ...<£ (2 ; «n, cn_2 ; an_l5 an)
has neither a zero nor an infinity at c1} at c2, ..., and at cn_2 ; it has simple
infinities at al} a2, ..., an, and simple zeros at alt a2, ..., an-1} an. Hence it
has the same irreducible infinities and the same irreducible zeros in the same
degree as the given function <£ (z) ; and therefore, by Prop. IV., <I> (z) is
a mere constant multiple of the foregoing product.

The theorem is thus completely proved.

Other developments for functions, the infinities of which are not simple,
are possible ; but they are relatively unimportant in view of a theorem,
Prop. XV., about to be proved, which expresses any periodic function in
terms of a single function of the second order and its derivative.

XIII. If two doubly -periodic functions have the same periods, they are
connected by an algebraical equation.

Let u be one of the functions, having n irreducible infinities, and v be
the other, having m irreducible infinities.

By Prop. VI., Corollary I., there are n irreducible values of z for a value
of u; and to each irreducible value of z there is a doubly-infinite series of.
values of z over the plane. The function v has the same value for all the
points in any one series, so that a single value of v can be associated uniquely
with each of the irreducible values of z, that is, there are n values of v for
each value of u. Hence, (§ 99), v is a root of an algebraical equation of the
nth degree, the coefficients of which are functions of u.

Similarly u is a root of an algebraical equation of the mth degree, the
coefficients of which are functions of v.

Hence, combining these results, we have an algebraical equation between
u and v of the nth degree in v and the mth in u, where m and n are the
respective orders of v and u.

COROLLARY I. If both the functions be even functions of z, then n and m
are even integers ; and the algebraical relation between u and v is of degree ^n
in v and of degree ^m in u.

COROLLARY II. If a function u be doubly-periodic in &> and &>', and a
function v be doubly -periodic in fl and U', where

n = mca + nta, I!' = m'w + nw! ,
m, n, m', n being integers, then there is an algebraic relation between u and v.

119. It has been proved that, if a doubly-periodic function u be of order m,
then its derivative du/dz is doubly-periodic in the same periods and is of an
order n, which is not less than m + 1 and not greater than 2?/i. Hence, by

119.] BETWEEN HOMOPERIODIC FUNCTIONS 237

Prop. XIII., there subsists between u and u an algebraical equation of order m
in u' and of order n in u; let it be arranged in powers of u' so that it takes
the form

U" u'm _j_ JJ u'm—i _i _ _ i U _ u'2 i JJ _u' i JJ __ Q

where U0, U1} ... , Um are rational integral algebraical functions of u one at
least of which must be of degree n.

Because the only distinct infinities of u' are infinities of u, it is impossible
that u' should become infinite for finite values of u: hence U0 = 0 can have no
finite roots for u, that is, it is a constant and so it may be taken as unity.

And because the m values of z, for which u assumes a given value, have
their sum constant save as to integral multiples of the periods, we have

corresponding to a variation 8u ; or

du du du

f/7/

Now — is one of the values of u' corresponding to the value of u, and so for

the others ; hence

3 1

r=i ur
that is, by the foregoing equation,

" m— i

= 0,

un.

and therefore Um-^ vanishes. Hence :

XIV. There is a relation, between a doubly -periodic function u of order m
and its derivative, of the form

u'm + U^'™-1 + ...+ U^u'* + Um = 0,

where Ul}..., Um_2, Um are rational integral algebraical functions of u, at
least one of which must be of degree n, the order of the derivative, and n is
not less than m + 1 and not greater than 2m.

Further, by taking v = - , which is a function of order m because it has the

Uj

m irreducible zeros of u for its infinities, and substituting, we have

vf™ _ 03 U^'m~l + v*U«v'm~* - . . . ± v2"1-4 Um_2v''2 + v2"1 Um = 0.
The coefficients of this equation must be integral functions of v ; hence the
degree of Ur in u cannot be greater than 2r.

COROLLARY. The foregoing equation becomes very simple in the case of
doubly-periodic functions of the second order.

Then m = 2.

238 DIFFERENTIAL EQUATION [119.

If the function have one infinity of the second degree, its derivative has
that infinity in the third degree, and is of the third order, so that n = 3 ; and
the equation is

/y7?/\2

( ^ ) = \u? + 3/iw2 + Svu + p,
\d*J

where X, /*, v, p are constants. If 6 be the infinity, so that

A
*.£(,)_-_— + £(*),

where % (^) is everywhere finite in the parallelogram, then - = ±A ; and the

/77/

zeros of -j- are 6 + o>, 0 + &/, 6 + o> + CD' ; so that
diz

a,')} {

This is £/ie general differential equation of Weierstrasss elliptic functions.

If the function have two simple infinities a and @, its derivative has each
of them as an infinity of the second degree, and is of the fourth order, so that
n = 4 ; and the equation is

(du\* _
(dz) =

dM + c2w + >c3u + c4,
where c0, c1} c2, c3, c4 are constants. Moreover

where ^ (^) is finite everywhere in the parallelogram. Then cu = G~2 ; and

^/'i/

the zeros of -y- are ^ (a + /3), -|- (a + (3) + w, ^ (a -f /3) + cof, % (a + ft) + w + &>',
ft/2

so that the equation is

(« + 13)+ « + «}].
This is the general differential equation of Jacobis elliptic functions.

The canonical forms of both of these equations will be obtained in Chapter
XI., where some properties of the functions are investigated as special illustra
tions of the general theorems.

Note. All the derivatives of a doubly-periodic function are doubly-
periodic in the same periods, and have the same infinities as the function but
in different degrees. In the case of a function of the second order, which
must satisfy one or other of the two foregoing equations, it is easy to see that
a derivative of even rank is a rational, integral, algebraical function of u, and
that a derivative of odd rank is the product of a rational, integral, algebraical
function of u by the first derivative of u.

119.] OF DOUBLY-PERIODIC FUNCTIONS 239

It may be remarked that the form of these equations confirms the result
at the end of § 117, by giving two values of u' for one value of u, the two
values being equal and opposite.

Ex. If u be a doubly-periodic function having a single irreducible infinity of the third
degree so as to be expressible in the form

2 6
— -o + -5 + integral function of z

z z

within the parallelogram of periods, then the differential equation of the first order which
determines u is

where £74 is a quartic function of u and where a is a constant which does not vanish with 6.

(Math. Trip., Part II, 1889.)

XV. Every doubly -periodic function can be expressed rationally in terms
of a function of the second order, doubly-periodic in the same periods, and its
derivative.

Let u be a function of the second order and the second class, having the
same two periods as v, a function of the rath order ; then, by Prop. XIII.,
there is an algebraical relation between u and v which, being of the second
degree in v and the mth degree in u, may be taken in the form

Lv* - 2Mv + P = 0,

where the quantities L, M, P are rational, integral, algebraical functions of u
and at least one of them is of degree m. Taking

Lv-M=w,

we have w2 = M'2 — LP,

a rational, integral, algebraical function of u of degree not higher than 2w.

Thus w cannot be infinite for any finite value of u : an infinite value of u
makes w infinite, of finite multiplicity. To each value of u there correspond
two values of w equal to one another but opposite in sign.

Moreover w, being equal to Lv - M , is a uniform function of z, say F(z\
while it is a two-valued function of u. A value of u gives two distinct
values of z, say zl and £2 ; hence the values of w, which arise from an assigned
value of u, are values of w arising as uniform functions of the two distinct
values of z. Hence as the two values of w are equal in magnitude and
opposite in sign, we have

r(4)+J*(4)-Oi

that is, since ^ + z.2 = a. + ft where a and /3 are the irreducible infinities of u,

so that l(a + £), £(a + /3) + a>, £(a + £) + «', and £ (a + /3) + a> + a>' are either
zeros or infinities of w. They are known not to be infinities of u, and w is
infinite only for infinite values of u ; hence the four points are zeros of w.

240 RELATIONS BETWEEN [119.

But these are all the irreducible zeros of u' ; hence the zeros of u' are
included among the zeros of w.

Now consider the function w/u'. The numerator has two values equal
and opposite for an assigned value of u ; so also has the denominator. Hence
w/u' is a uniform function of u.

This uniform function of u may become infinite for
(i) infinities of the numerator,
(ii) zeros of the denominator.

But, so far as concerns (ii), we know that the four irreducible zeros of the
denominator are all simple zeros of u' and each of them is a zero of w .; hence
w/u' does not become infinite for any of the points in (ii). And, so far as
concerns (i), we know that all of them are infinities of u. Hence w/u, a
uniform function of u, can become infinite only for an infinite value of u, and
its multiplicity for such a value is finite; hence it is a rational, integral,
algebraical function of u, say N, so that

w = Nu'.

Moreover, because w2 is of degree in u not higher than 2m, and u'2 is of
the fourth degree in u, it follows that N is of degree not higher than m — 2.

We thus have Lv — M — Nu,

M+Nu M N ,

v= ~r = L + LU>

where L, M, N are rational, integral, algebraical functions of u ; the degrees
of L and M are not higher than m, and that of N is not higher than m — 2.

Note 1. The function u, which has been considered in the preceding
proof, is of the second order and the second class. If a function u of the
second order and the first class, having a double irreducible infinity, be
chosen, the course of proof is similar ; the function w has the three irreducible
zeros of u' among its zeros and the result, as before, is

w = Nu'.

But, now, w"- is of degree in u not higher than 2m and u'2 is of the third
degree in u ; hence N is of degree not higher than m — 2 and the degree of w2
in u cannot be higher than 2m — 1.

Hence, if L, M, P be all of degree m, the terms of degree 2m in LP — M2
disappear. If all of them be not of degree m, the degree of M must not be
higher than m — l ; the degree of either L or P must be m, but the degree
of the other must not be greater than m—l, for otherwise the algebraical
equation between u and v would not be of degree m in u.

We thus have

Lv2 - 2Mv + P = (), Lv - M = Nu',

119.] HOMOPERIODIC FUNCTIONS 241

where the degree of N in u is not higher than m — 2. If the degree of L be
less than TO, the degree of M is not higher than TO — 1 and the degree of P is
TO. If the degree of L be m, the degree of M may also be m provided that the
degree of P be TO and that the highest terms be such that the coefficient
of u2m in LP - M'2 vanishes.

Note 2. The theorem expresses a function v rationally in terms of u and
u : but u' is an irrational function of u, so that v is not expressed rationally
in terms of u alone.

But, in Propositions XI. and XII., it was indicated that a function such as
v could be rationally expressed in terms of a doubly-periodic function, such as
u. The apparent contradiction is explained by the fact that, in the earlier
propositions, the arguments of the function u in the rational expression and
of the function v are not the same ; whereas, in the later proposition whereby
v is expressed in general irrationally in terms of u, the arguments are the
same. The transition from the first (which is the less useful form) to the
second is made by expressing the functions of those different arguments in
terms of functions of the same argument when (as will appear subsequently, in
§ 121, in proving the so-called addition-theorem) the irrational function of u,
represented by the derivative u, is introduced.

COROLLARY I. Let H denote the sum of the irreducible infinities or of
the irreducible zeros of the function u of the second order, so that H = 2y for
functions of the first class, and O = a + /3 for functions of the second class.
Let u be represented by <f> (z) and v by ty (z), when the argument must be put
in evidence. Then

so that W-Z) =

J-j i_j ±j

Hence ^ (z) + ^ (fl - z) = 2 ^= 2R,

JL

First, if y (z) = ,Jr (ft - z\ then S = 0 and ^ (z) = R : that is, a function ^ (z),
which satisfies the equation

can be expressed as a rational algebraical meromorphic function of <f> (z) of the
second order, doubly -periodic in the same periods and having the sum of its
irreducible infinities congruent with O.

Second, if ^ (e) = - y, (fl _ z\ then R = 0 and ^ (*) = flf (*) ; that is,
function ^ (z), which satisfies the equation

16

a

242 HOMOPERIODIC FUNCTIONS [119.

can be expressed as a rational algebraical meromorphic function of <£ (z),
multiplied by 0' (z), where $ (z} is doubly-periodic in the same periods, is of the
second order, and has the sum of its irreducible infinities congruent with Q.

Third, if ty(z) have no infinities except those of u, it cannot become
infinite for finite values of u ; hence L = 0 has no roots, that is, L is a constant
which may be taken to be unity. Then i/r (z) a function of order m can be
expressed in the form

where, if the function </> (z) be of the second class, the degree of M is not
higher than m ; but, if it be of the first class, the degree of M is not higher
than m - 1 ; and in each case the degree of N is not higher than m - 2.

It will be found in practice, with functions of the first class, that these
upper limits for degrees can be considerably reduced by counting the degrees
of the infinities in

Thus, if the degree of M in u be ^ and of N be \ the highest degree of an :
infinity is either 2/t or 2X + 3 ; so that, if the order of ^ (z) be m, we should

have

m = 2/j, or m = 2\ + 3, >

according as m is even or odd.

When functions of the second class are used to represent a function ^r (z),
which has two infinities a and /3 each of degree n, then it is easy to see that
M is of degree n and N of degree n - 2 ; and so for other cases.

COROLLARY II. Any doubly -periodic function can be expressed rationally
in terms of any other function u of any order n, doubly-periodic in the same
periods, and of its derivative ; and this rational expression can always be taken
in the form

U0 + U,U' + t/X3 + • • • + Un-,u'n~\

where U0, ... , £7n-i are algebraical, rational, meromorphic functions of u.

COROLLARY III. If <f) be a doubly-periodic function, then <f> (u + v) can be
expressed in the form

where ^ is a doubly -periodic function in the same periods and of the second
order : each of the functions A, D, E is a symmetric function of^(u) and i/r (v),
and B is the same function of^(v) and ty(u) as C is of ty (u) and ty (v).

The degrees of A and E are not greater than m in ty (u) and than m in ^ (v),
where m is the order of </> ; the degree of D is not greater than m - 2 in ^ (u)
and than m - 2 in ^ (v) ; the degree of B is not greater than m - 2 in ^ (u)
and than m in ^ (v), and the degree of C is not greater than m - 2 in -^ (v)
and than m in -^ (u).

CHAPTER XI.

DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER.

THE present chapter will be devoted, in illustration of the preceding
theorems, to the establishment of some of the fundamental formulae relating
to doubly-periodic functions of the second order which, as has already (in
§ 119, Cor. to Prop. XIV.) been indicated, are substantially elliptic functions :
but for any development of their properties, recourse must be had to treatises
on elliptic functions.

It may be remarked that, in dealing with doubly-periodic functions, we
may restrict ourselves to a discussion of even functions and of odd functions.
For, if (/> (z) be any function, then £ {<j> (z} + <j>(— z}} is an even function, and
\ {(f>(z) — </>(— z}} is an odd function, both of them being doubly-periodic in
the periods of <f> (z) ; and the new functions would, in general, be of order
double that of <J>(z). We shall practically limit the discussion to even
functions and odd functions of the second order.

120. Consider a function <j>(z\ doubly-periodic in 2&> and 2w'; and let
it be an odd function of the second class, with a and ft as its irreducible
infinities, and a and b as its irreducible zeros*.

Then we have <£ (z) = (f> (a + /3 — z)

which always holds, and <f> (— z) = — </> (z)

which holds because <£ (z) is an odd function. Hence

<f> (a + /3 + z) = (/>(- *)

= -$(*)

so that a + ft is not a period ; and

-*(*),

To fix the ideas, it will be convenient to compare it with snz, for which 2w = 4^T, 2<a' =
a=iK', p=iK' + 2K, a-0, and b = 2K.

16—2

244 DOUBLY-PERIODIC FUNCTIONS [120.

whence 2 (a + /S) is a period. Since a -f /3 is not a period, we take a + /3 = a>,
or = &)', or = &> + w' ; the first two alternatives merely interchange &> and &>', so

that we have either

a + /3 = o),

or a + /3 = ft) + &>'.

And we know that, in general,

a + b = a + /3.
First, for the zeros : we have

so that </>(0) is either zero or infinite. The choice is at our disposal; for

- satisfies all the equations which have been satisfied by $(z) and an

</>(*)

infinity of either is a zero of the other. We therefore take

so that we have a = 0,

6 = to or &) + ft)'.
Next, for the infinities : we have

*(*)—$(-*)

and therefore <j> (- a) = - $ (a) = oo .

The only infinities of <£ are a and /3, so that either

— a= a,

or -CL = P.

The latter cannot hold, because it would give a + /3 = 0 whereas

or = &> + &/; hence

2a = 0,

which must be associated with a + /3 = w or with a + /3 = &> + &/.

Hence a, being a point inside the fundamental parallelogram, is either 0,
a), &)', or tw + &)'.

It cannot be 0 in any case, for that is a zero.

If a _|_ ^ = Wj then a cannot be tw, because that value would give ft = 0,
which is a zero, not an infinity. Hence either a = «', and then /3 = &/ + &>;
or a = &)' + &), and then /3 = ft)'. These are effectively one solution ; so that, if

a + /3 = &), we have

a, /3 = ft)', &>' + &))

and a, 6 = 0, &) ) '

jf a + /S = w + &>', then a cannot be CD f &)', because that value would give
{$ = 0, which is a zero, not an infinity. Hence either a = &> and then ft = &)',
or a = ft)' and then /3 = &). These again are effectively one solution ; so that,

if a + /3 = &) + &>', we have

a, £ = o), ft)'
and a, 6 = 0, «o + ft)')

120.] OF THE SECOND CLASS 245

This combination can, by a change of fundamental parallelogram, be made
the same as the former ; for, taking as new periods

2ft/ = 2a>'t 2fl = 2« + 2a>',
which give a new fundamental parallelogram, we have a + j3 = H, and

a, ft — &>', ft — ft/, that is, ft/, ft — 03' + 2<o'
so that a, /3 = ft/, O + a/]

and a, b = 0,

being the same as the former with O instead of &>. Hence it is sufficient to
retain the first solution alone : and therefore

a = to', ft = CD' + co,
a = 0, 6 = w.

Hence, by § 116, 1., we have

where F(z) is finite everywhere within the parallelogram.

Again, $ (z + a/) has z = 0 and z = &> as its irreducible infinities, and
it has 2 = 0)' and z = &> + &/ as its irreducible zeros, within the parallelogram
of (f) (z} ; hence

where ^ (2) is finite everywhere within the parallelogram. Thus

a function which is finite everywhere within the parallelogram ; since it is
doubly-periodic, it is finite everywhere in the plane and it is therefore a
constant and equal to the value at any point. Taking - i&/ as the point
(which is neither a zero nor an infinity) and remembering that </> is an odd
function, we have

* (*)*(* + «0 = - ft (*»')}' = p

k being a constant used to represent the value of - {<£ (^o/)}"2.

Also <j>(z + o)) = <f>(z + a + /3- 2&/)

= c/>0 + a + /3)=-(£ (z),
and therefore also <£ (&> — z) = <f) (z).

The irreducible zeros of <j>' (z) were obtained in § 117, X. In the
present example, those points are a>' + £ft>, &>' + ffc>, £&>, f &> ; so that, as
there, we have

£to'('W-{*(i)-4>(i*yito(*)-HW

where K is a constant. But

$ (f®) = 0 (2® - lft>) = (f) (-!«) = _(£ (1 a,) ;

246 DOUBLY-PERIODIC FUNCTIONS [120.

and 0(fw + &/) = <£(2a> + 2w' -.!&>- a/)

= <£(- 2 <o -to')

= — </>(£&> + &>');

so that • ,. . 4 I -

where J. is a new constant, evidently equal to {<£'(0)}2. Now, as we know
the periods, the irreducible zeros and the irreducible infinities of the function
</> (z), it is completely determinate save as to a constant factor. To determine
this factor we need only know the value of <$>(z) for any particular finite
value of z. Let the factor be determined by the condition

then, since <£(^ft>)<£(^G> + ft/) = T

by a preceding equation, we have

and then

ft' (*)}» - {f (0)}« [1 - {<£ (*)}2] [1 - fr {(/> (*)}']

Hence, since (/> (2) is an odd function, we have

<£ (z) = sn (//,£).

Evidently 2/xtu, 2/^ft)' = 4^T, 2^', where K and ^T' have the ordinary signifi
cations. The simplest case arises when /A = 1.

121. Before proceeding to the deduction of the properties of even
functions of z which are doubly-periodic, it is desirable to obtain the
addition-theorem for <f>, that is, the expression of <p (y + z) in terms of
functions of y alone and z alone.

When <f> (y + z) is regarded as a function of z, which is necessarily of the
second order, it is (§ 119, XV.) of the form

where M and L are of degree in <£ (z) not higher than 2 and N is independent
of z. Moreover y + z = a and y + z = ft are the irreducible simple infinities
of <j) (y + z) ; so that L, as a function of z, may be expressed in the form

and therefore

Z±_^(iHL^^^)}l
(z) -

121.] OF THE SECOND CLASS 247

where P, Q, R, S are independent of z but they may be functions of y. Now

</> (a - y) = </> (w' - y) = -

and <£ (/3 — y) = <j> (&>' + w — y} =

so that the denominator of the expression for <f> (y + 2) is

Since </> (z) is an odd function, <£' (#) is even ; hence

,A p -
~ */

and therefore $ (y + z) — $ (y — z) = - -

Differentiating with regard to z and then making z = 0, we have

so that, substituting for Q we have

Interchanging y and z and noting that </> (t/ — z) = — (f) (z — y), we have

md therefore d> C7y * Z} d>' (0} -

W+*)1><

which is the addition-theorem required.

Ex. If f(u) be a doubly-periodic function of the second order with infinities 61} i2,
and 0(tt) a doubly -periodic function of the second order with infinities alt a2 such that,
in the vicinity of «» (for i — 1, 2), we have

^ (M) = ,7~!r +Pi+& (u~ai) + ...... >

c6 — u-j

thon /M-/W = • i» W+* W-ft-ftl-

the periods being the same for both functions. Verify the theorem when the functions are
sn u and sn (u + v}. (Math. Trip. Part II., 1 891.)

Prove also that, for the function $ (u), the coefficients p± and p2 are equal. (Burnside.)

122. The preceding discussion of uneven doubly-periodic functions
having two simple irreducible infinities is a sufficient illustration of the

248 DOUBLY-PERIODIC FUNCTIONS [122.

method of procedure. That, which now follows, relates to doubly- periodic
functions with one irreducible infinity of the second degree ; and it will be
used to deduce some of the leading properties of Weierstrass's er-function
(of § 57) and of functions which arise from it.

The definition of the <r-function is

where fi = 2ma> + 2m'a)', the ratio of &>' : &> not being purely real, and the
infinite product is extended over all terms that are given by assigning to
m and to m' all positive and negative integral values from +00 to — oo ,
excepting only simultaneous zero values. It has been proved (and it is
easy to verify quite independently) 'that, when cr(z) is regarded as the
product of the primary factors

the doubly-infinite product converges uniformly and unconditionally for all
values of z in the finite part of the plane ; therefore the function which it
represents can, in the vicinity of any point c in the plane, be expanded in a
converging series of positive powers of z — c, but the series will only express
the function in the domain of c. The series, however, can be continued over
the whole plane.

It is at once evident that a- (z) is not a doubly-periodic function, for it has
no infinity in any finite part of the plane.

It is also evident that a (z) is an odd function. For a change of sign in z
in a primary factor only interchanges that factor with the one which has
equal and opposite values of m and of m', so that the product of the two factors
is unaltered. Hence the product of all the primary factors, being independent
of the nature of the infinite limits, is an even function ; when z is associated
as a factor, the function becomes uneven and it is a- (z).

The first derivative, a' (z), is therefore an even function ; and it is not
infinite for any point in the finite part of the plane.

It will appear that, though a- (z) is not periodic, it is connected with
functions that have 2o> and 2&>' for periods ; and therefore the plane will be
divided up into parallelograms. When the whole plane is divided up, as in
§ 105, into parallelograms, the adjacent sides of which are vectorial repre
sentations of 2w and 2&/, the function a-(z) has one, and only one, zero in
each parallelogram; each such zero is simple, and their aggregate is given
by z = £l. The parallelogram of reference can be chosen so that a zero
of <r (z} does not lie upon its boundary ; and, except where explicit account is

122.] OF THE FIRST CLASS 249

taken of the alternative, we shall assume that the argument of &>' is greater
than the argument of to, so that the real part* of w'/ia) is positive.

123. We now proceed to obtain other expressions for a- (z), and particu
larly, in the knowledge that it can be represented by a converging series in
the vicinity of any point, to obtain a useful expression in the form of a series,
converging in the vicinity of the origin.

Since er (z) is represented by an infinite product that converges uniformly
and unconditionally for all finite values of z, its logarithm is equal to the sum
of the logarithms of its factors, so that

where the series on the right-hand side extends to the same combinations of
m and m' as the infinite product for z, and, when it is regarded as a sum of

z z^ ( z\

functions o + i 7^2 + ^°£ ( ^ ~ r> ) » ^ne sei>ies converges uniformly and uncon-

__ -- \ 1 - ,

ditionally, except for points z = £l. This expression is valid for log a (z) over
the whole plane.

Now let these additive functions be expanded, as in § 82. In the imme
diate vicinity of the origin, we have

a series which converges uniformly and unconditionally in that vicinity.
Then the double series in the expression for log a (z} becomes

and as this new series converges uniformly and unconditionally for points in
the vicinity of z = 0, we can, as in § 82, take it in the form

oo ~r ( oo oo }

5" J 5* y O-n

— 4 — \ ^-> <5r »• ()

r=3 r (-00 -oo J

which will also, for such values of z, converge uniformly and unconditionally.
In § 56, it was proved that each of the coefficients

00 00

2 s n-*-,

— 00 - 00

for r = 3, 4,..., is finite, and has a value independent of the nature of the
infinite limits in the summation. When we make the positive infinite limit
for m numerically equal to the negative infinite limit for m, and likewise for

This quantity is often denoted by ffi ( . - J .

250 WEIERSTRASS'S [123.

ra', then each of these coefficients determined by an odd index r vanishes,
and therefore it vanishes in general. We then have

log a- 0) = log z - I* 22ft-4 - ^ 22ft-6 - ^ 22ft-8

a series which converges uniformly and unconditionally in the vicinity of the
origin.

The coefficients, which occur, involve «o and «', two independent constants.
It is convenient to introduce two other magnitudes, g.2 and g3, denned by the

equations

#2= 6022ft-4, #3 = 140220-0,

so that g2 and </3 are evidently independent of one another; then all the
remaining coefficients are functions* of g.2 and g3. We thus have

and therefore <r (z) = ze m

where the series in the index, containing only even powers of z, converges

uniformly and unconditionally in the vicinity of the origin.

It is sufficiently evident that this expression for a- (z) is an effective
representation only in the vicinity of the origin ; for points in the vicinity of
any other zero of cr (z), say c, a similar expression in powers of z - c instead
of in powers of z would be obtained.

124. From the first form of the expression for log cr (z), we have

o-(z) z _«, _

where the quantity in the bracket on the right-hand side is to be regarded as
an element of summation, being derived from the primary factor in the
product-expression for cr (z\

We write £(z) = , ^ ,

so that %(z) is, by § 122, an odd function, a result also easily derived from the
foregoing equation ; and so

This expression for £ (z) is valid over the whole plane.
Evidently £ (z) has simple infinities given by

for all values of ra and of m between + oo and - oo , including simultaneous
zeros. There is only one infinity in each parallelogram, and it is simple ; for
the function is the logarithmic derivative of a (z\ which has no infinity and

* See Quart. Journ., vol. xxii., pp. 4, 5. The magnitudes g2 and g3 are often called the
invariants.

124.] ELLIPTIC FUNCTION 251

only one zero (a simple zero) in the parallelogram. Hence %(z) is not a
doubly-periodic function.

For points, which are in the immediate vicinity of the origin, we have

but, as in the case of cr(z), this is an effective representation of %(z) only
in the vicinity of the origin ; and a different expression would be used for
points in the vicinity of any other infinity.

We again introduce a new function g> (z) defined by the equation

Because £ is an odd function, $ (z) is an even function ; and

where the quantity in the bracket is to be regarded as an element of
summation. This expression for $ (z) is valid over the whole plane.
Evidently |p (z) has infinities, each of the second degree, given by z = fl,
for all values of m and of m between -f oo and - oo , including simultaneous
zeros ; and there is one, and only one, of these infinities in each parallelogram.
One of these infinities is the origin; using the expression which represents
log a- (z) in the immediate vicinity of the origin, we have

= -2 + 20 9** + ^ 9*?+ • • •

for points z in the immediate vicinity of the origin. A corresponding
expression exists for g> (z) in the vicinity of any other infinity.

125. The importance of the function $ (z) is due to the following
theorem : —

The function $> (z) is doubly-periodic, the periods being 2<w and 2&/.
Wo have -l

where the doubly-infinite summation excludes simultaneous zero values, and
the expression is valid over the whole plane. Hence

+ ^-n - Si

252 WEIERSTRASS'S [125.

so that

obtained by combining together the elements of the summation in g> (z + 2<w)
and |p (,z). The two terms, not included in the summation, can be included,
if we remove the numerical restriction as to non-admittance of simultaneous
zero values for m and m'\ and then

2.) - f (,) = 2

_

where now the summation is for all values of m and of m' from + oo to — oo .
Let q denote the infinite limit of m, and p that of m'. Then terms in the
first fraction, for 0 = 2 (mm + m'w'}, are the same as terms in the second for
£1 = 2 (m — l)w + 2m' w ; cancelling these, we have

m'=p

-fC =

where q is infinite. But

?r)2 sin2 c '
and therefore

»»' = p = oo 1 ^.2

2i

- 2mV}2 W .
sin

2o/

if/) be infinitely great compared with q. This condition may be assumed for
the present purpose, because the value of g> (z) is independent of the nature
of the infinite limits in the summation and is therefore unaffected by such a
limitation.

f - "" 1 ] f £+!?(9+1) -*$-* *

l_* J ' l_

The fraction —, has a real part. In the exponent it is multiplied by q + 1.

that is, by an infinite quantity ; so that the real part of the index of
the exponential is infinite, either positive or negative. Thus either the
first term is infinite and the second zero, or vice versa; in either case,

r T i •

sin \z + 2 (q + 1) twl ^— , is infinite, and therefore
2o) J

{2 + 2(q + l)(o- 2m V}2
Similarly for the other sum. Hence

= 0.

In the same way it may be shewn that

£>0 + 2a/)-£>0) = 0;

therefore £> (z) is doubly-periodic in 2<o and 2a>'.

126.] ELLIPTIC FUNCTION 253

Now in any parallelogram whose adjacent sides are 2&> and 2&>', there is
only one infinity and it is of multiplicity two; hence, by § 116, Prop. III.,
Cor. 3, 2o) and 2&>' determine a primitive parallelogram for $> (z).

We shall assume the parallelogram of reference chosen so as to include
the origin.

126. The function $ (z) is thus of the second order and the first class.

Since its irreducible infinity is of the second degree, the only irreducible
infinity of g>' (z} is of the third degree, being the origin ; and the function
<§t ' (z) is odd.

The zeros of jp' (z} are thus &>, ft/, and (&> + to') ; or, if we introduce a new
quantity w" defined by the equation

&>" = &) + &>',

the zeros of <@! (z) are &>, &>', &>".
We take

#>(«) = e1} p(a>") = e2, p(m') = e3, %>(z) = p:
and then, by § 119, Prop. XIV., Cor., we have

where A is some constant. To determine the equation more exactly, we
substitute the expression of jp in the vicinity of the origin. Then

80 that P' = -j+iQff* +

When substitution is made, it is necessary to retain in the expansion all
terms up to z° inclusive. We then have, for |p'2, the expression

4 2 4

and for A (^ - e-,) (p - e2) (p - e3), the expression

A r1 3 9- 3

1L^6 + 20^+285r3+>"

- (e, + e2 + e3)(^+—g2 + ...)+ (6le, + e2e3 + e&) (- + ...)- tf,«A |

When we equate coefficients in these two expressions, we find

e1 + e2 + e3 = 0, e& + e»
therefore the differential equation satisfied by p is

254 PERIODICITY [126.

Evidently £>" = 6§>2 - %gs,

and so on ; and it is easy to verify that the 2wth derivative of g) is a rational
integral algebraical function of <p of degree n + l and that the (2w+l)th
derivative of fp is the product of g>' by a rational integral algebraical function
of degree n.

The differential equation can be otherwise obtained, by dependence on
Cor. 2, Prop. V. of § 116. We have, by differentiation of %>',

for points in the vicinity of the origin ; and also

^+!^2 + r4^2 + "--

Hence <@" and §>2 have the same irreducible infinities in the same degree and
their fractional parts are essentially the same : they are homoperiodic and
therefore they are equivalent to one another. It is easy to see that g>" — 6(jp2
is equal to a function which, being finite in the vicinity of the origin, is finite
in the parallelogram of reference and therefore, as it is doubly -periodic, is
finite over the whole plane. It therefore has a constant value, which can be
obtained by taking the value at any point; the value of the function for

z = 0 is — \g» and therefore

g>"_6^ = -^2,

so that |p"= 6g»2-|<72,

the integration of which, with determination of the constant of integration,

leads to the former equation.

This form, involving the second derivative, is a convenient one by which
to determine a few more terms of the expansion in the vicinity of the origin :
and it is easy to shew that

from which some theorems relating to the sums ^SH"2*1 can be deduced*.

Ex. If cn be the coefficient of 22n~2 in the expansion of $ (z) in the vicinity of the
origin, then

c»=/o~ . iw.. ON 2 Crfin-r- (Weierstrass.)

We have jp'2 = 4^>3 - g$ - g3 ;

the function jjp' is odd and in the vicinity of the origin we have

* See a paper by the author, Quart. Journ., vol. xxii, (1887), pp. 1 — 43, where other references
are given and other applications of the general theorems are made.

126.] OF WEIERSTRASS'S FUNCTION 255

hence, representing by — (4|p3 — g$>— g^ that branch of the function which is
negative for large real values, we have

and therefore z =

The upper limit is determined by the fact that when z = 0, g> = oo ; so that

- r d®

_ r d%>

lp {4 (p - ej (p - e2) (p - e,)}*'

This is, as it should be, an integral with a doubly-infinite series of values.
We have, by Ex. 6 of § 104,

r

0)j = ft) =|

J<h

,

ft>3= ft) =

J

with the relation a)" = a) + co'.

127. We have seen that g> (z) is doubly-periodic, so that

p(*+2») = $>(*),

and therefore dg(5 + 2«) = dgW

a^ dz

hence integrating ?(^ + 2<») = %(z) + A.

Now ^ is an odd function ; hence, taking z — — co which is not an infinity of £,
we have

^ = 2^(&))=27;

say, where r) denotes £ (&>) ; and therefore

£(*-*• 2»)r- £•(*)« 89,

which is a constant.

Similarly %(z + 2&>') - ^ (^) - 2i/,

where r;' = ^ (to') and is constant.

Hence combining the results, we have

% (z + 2w&) + 2?rc V) -£(z) = 2mri + Zmrj',
where m and m are any integers.

It is evident that 77 and rj' cannot be absorbed into £; so that £ is not a
periodic function, a result confirmatory of the statement in § 124.

256 PSEUDO-PERIODIC [127.

There is, however, a pseudo-periodicity of the function £ : its characteristic
is the reproduction of the function with an added constant for an added
period. This form is only one of several simple forms of pseudo-periodicity
which will be considered in the next chapter.

128. But, though %(z) is not periodic, functions which are periodic can
be constructed by its means.

Thus, if 4>(z)=AS(z-a)+Bt(z-V)
then * + 2w-(*) = 2A£(*-a

so that, subject to the condition

A+B+C+...=0,
<j) (z) is a doubly-periodic function.

Again, we know that, within the fundamental parallelograTH, f has a
single irreducible infinity and that the infinity is simple; hence the irre
ducible infinities of the function </> (V) are z = a, b, c, ..., and each is a simple
infinity. The condition A + B + C + ...=0 is merely the condition of Prop.
III., § 116, that the ' integral residue ' of the function is zero.

Conversely, a doubly-periodic function with m assigned infinities can be
expressed in terms of f and its derivatives. Let ax be an irreducible infinity
of <£> of degree n, and suppose that the fractional part of <I> for expansion in
the immediate vicinity of ax is

A! i?i | ^ KI

Then

-if (*- 4).-...

is not infinite for z = a^.

Proceeding similarly for each of the irreducible infinities, we have a
function

r

which is not infinite for any of the points z = alt a2, .... But because <f> (z)
is doubly-periodic, we have

and therefore the function

128.]

FUNCTIONS

257

is doubly-periodic. Moreover, all the derivatives of any order of each of the
functions £ are doubly-periodic; hence the foregoing function is doubly-
periodic.

The function has been shewn to be not infinite at the points a1} a2, ...,
and therefore it has no infinities in the fundamental parallelogram ; con
sequently, being doubly-periodic, it has no infinities in the plane and it is
a constant, say G. Hence we have

g,

r=i

r=i

m

with the condition 2 Ar = 0, which is satisfied because <E> (z) is doubly-
periodic.

This is the required expression * for <I> (z) in terms of the function % and
its derivatives; it is evidently of especial importance when the indefinite
integral of a doubly-periodic function is required.

129. Constants 77 and 77', connected with &> and «', have been introduced
by the pseudo-periodicity of £(z)\ the relation, contained in the following
proposition, is necessary and useful : —

The constants 77, w', &>, &>' are connected by the relation

the + or - sign being taken according as the real part of o>'fa)i is positive or
negative.

A fundamental parallelogram having an angular point at z0 is either of

the form (i) in fig. 34, in which case 9t f-^] is

\mj

positive : or of the form (ii), in which case 9J ( — . )

\Ct)l/

is negative. Evidently a description of the paral
lelogram A BCD in (i) will give for an integral the
'same result (but with an opposite sign) as a de
scription of the parallelogram in (ii) for the same
integral in the direction A BCD in that figure.

We choose the fundamental parallelogram, so
that it may contain the origin in the included
area. The origin is the only infinity of £ which
can be within the area : along the boundary £ is
always finite.

Now since

* See Hermitet Ann. de Toulouse, t. ii, (1888), C, pp. 1—12.
F.

20+2o)'

Fig. 34

258 PSEUDO-PERIODICITY OF WEIERSTRASS's [129.

the integral of £(z) round ABCD in (i), fig. 34, is (§ 116, Prop. II., Cor.)

rD CB

2r)dz - fy'dz,

J A J A

the integrals being along the lines AD and AB respectively, that is, the

integral is

4 (rju>' — rfw}.

But as the origin is the only infinity within the parallelogram, the path of
integration ABCD A can be deformed so as to be merely a small curve round !
the origin. In the vicinity of the origin, we have

and therefore, as the integrals of all terms except the first vanish when taken
round this curve, we have

= 2-Trt.

Hence 4 (rjw — TJ'O)) = 27ri,

and therefore i](f> — rju> = \iri.

This is the result as derived from (i), fig. 34, that is, when 91 [?-) is positive.

\i/tU/

When (ii), fig. 34, is taken account of, the result is the same except
that, when the circuit passes from z0 to z0 + 2&>, then to z0 + 2t» + 2o>',
then to z0 + 2&>' and then to z0, it passes in the negative direction round the'
parallelogram. The value of the integral along the path ABCDA is the
same as before, viz., 4 (rjw — rj'a)) ; when the path is deformed into a small!

rdz

curve round the origin, the value of the integral is I — taken negatively, an

J **

therefore it is — 2?ri : hence

t](£) — rj (a = — \Tri.

Combining the results, we have

rjay' — f]w = ± ^Tri,
/ '\
according: as 9t ( — . ] is positive or negative.

\0)lJ

COROLLARY. If there be a change to any other fundamental parallelo
gram, determined by 2H and 2O', where

£1 = pa) + qa)', £1' = p'co + q'a)',
p, q, p', q' being integers such that pq — p'q = ± 1, and if H, H' denote

C(ft'), then

H = pr} + qrj', H' = p'rj + q'f}' ;

therefore HW - H'£l = ± \-iri,

according as the real part of T^ is positive or negative.

130.] PRODUCT-FUNCTIONS 259

130. It has been seen that £ (z) is pseudo-periodic ; there is also a pseudo-
periodicity for o- (z), but of a different kind. We have

that is,

0-0 +

and therefore a- (z + 2<w) = Ae^zcr (z),

where A is a constant. To determine A, we make z = — &>, which is not a
zero or an infinity of a (z) ; then, since a (z) is an odd function, we have

so that o- 0 + 2&>) = - e*> <z+<0> a- (z).

Hence o-(z + 4eo) = — e*> (z+3ft)) <r(z + 2&>)

and similarly a (z + 2mey) = (— l)m

Proceeding in the same way from

we find a~(z + 2m V) = (- l)m' e*>' <w/z+m'2w') o- (z).

Then a (z + 2ma> + 2m V) = (- l)m e2^ (ww+^»+»»»»V) Q- (^ + 2m/eo/)

== / _ J \m+m' g sz (mij+m'V) +2'?m2<o+47]mmV+27)'m'2co' _.

But lyct)' — r/o) = ± \iri,

SO that g2mm'(r|a>'— rj'w) _ e±mm'iri _ /_ |\nj.m'

and therefore

2m V) = (- l)w

which is the law of change of a (z) for increase of z by integral multiples of
the periods.

Evidently <r(z) is not a periodic function, a result confirmatory of the
statement in § 122. But there is a pseudo-periodicity the characteristic of
which is the reproduction, for an added period, of the function with an
exponential factor the index being linear in the variable. This is another
of the forms of pseudo-periodicity which will be considered in the next
chapter.

131. But though <r(z) is not periodic, we can by its means construct
functions which are periodic in the pseudo-periods of a (z).

By the result in the last section, we have

<r (z — a. + 2ma> + 2m'&)') cr (z — a) + ,,
<r(z-fi + 2mo> + 2m V) ~ <r(z ^~J3) & '

17—2

260 DOUBLY-PERIODIC FUNCTIONS [131.

and therefore, if <f> (z) denote

a- (z — cti) a (z — 02) cr (z — &»)

then $ (z + 2ra&> + 2m V) = e2(m>} +m'*'> <2^-2^

so that $ (z) is doubly-periodic in 2« and 2&>' provided

Now the zeros of <f>(z), regarded as a product of o--functions, are als a2,..., «„
and the points homologous with them ; and the infinities are Pi, /32, ... , ftn and i
the points homologous with them. It may happen that the points a and ft j
are not all in the parallelogram of reference ; if the irreducible points
homologous with them be a1} ..., an and blt ... , bn, then

Sar = ~S.br (mod. 2&>, 2co'),

and the new points are the irreducible zeros and the irreducible infinities of
<}>(z). This result, we know from Prop. III., § 116, must be satisfied.

It is naturally assumed that no one of the points a is the same as, or is
homologous with, any one of the points ft : the order of the doubly-periodic
function would otherwise be diminished by 1.

If any a be repeated, then that point is a repeated zero of <j>(z); similarly-
if any ft be repeated, then that point is a repeated infinity of <£ (z). In every,
case, the sum of the irreducible zeros must be congruent with the sum of the
irreducible infinities in order that the above expression for <j)(z) may be
doubly-periodic.

Conversely, if a doubly-periodic function <£ (z) be required with m assignedJ
irreducible zeros a and m assigned irreducible infinities b, which are subject t

to the congruence

2a = 26 (mod. 2co, 2&>'),

we first find points OL and ft homologous with a and with b respectively sucht
that

rru +U t «

Then the function

a- (z - Pi) a(z — ftm)

has the same zeros and the same infinities as </> (z), and is homoperiodic withi
it ; and therefore, by § 116, IV.,

o-(s-ai) o-^-otm)

9 \z> — •"• „(„ _ o \ *(*—ft V

where A is a quantity independent of z.

Ex. 1. Consider ft? (z). It has the origin for an infinity of the third degree and all thti
remaining infinities are reducible to the origin ; and its three irreducible zeros are a, a/, a>" j
Moreover, since <o"=a>' + a>, we have w + w' + w" congruent with but not equal to zerw
We therefore choose other points so that the sum of the zeros may be actually the same,

131.] EXAMPLES 261

as the sum of the infinities, which is zero ; the simplest choice is to take <», &>', - «".
Hence

where A is a constant. To determine A, consider the expansions in the immediate
vicinity of the origin ; then

2 o- ( - co) <r ( - to ) v (a)")

?"•" ...... S3 * ...... >

sothat y^-g^^/rf^'tf.

O- («) or (eo ) o- (a ) O"3 (2)

Another method of arranging zeros, so that their sum is equal to that of the infinities,
is to take — w, — «', co" ; and then we should find

dy M =2
r W

This result can, however, be deduced from the preceding form merely by changing the
sign of z.

Ex. 2. Consider the function

. a- (u + v) a- (u — v)

*«(«)

where v is any quantity and A is independent of u. It is, qua function of u, doubly-
periodic ; and it has u = 0 as an infinity of the second degree, all the infinities being
homologous with the origin. Hence the function is homoperiodic with g> (u) and it has
the same infinities as $> (u) : thus the two are equivalent, so that

where B and C are independent of u. The left-hand side vanishes if n—v; hence
(v), and therefore

where A' is a new quantity independent of u. To determine .4' we consider the
expansions in the vicinity of u = 0 ; we have

A.'<r(v)<r(-v)

sothat

and therefore cr- =

o-2 (%) o-2 (v)

a formula of very great importance.

Ex. 3. Taking logarithmic derivatives with regard to u of the two sides of the last
equation, we have

and, similarly, taking them with regard to D, we have

whence

262 EXAMPLES [131.

giving the special value of the left-hand side as (§ 128) a doubly-periodic function. It is
also the addition-theorem, so far as there is an addition-theorem, for the ^-function.

Ex. 4. We can, by differentiation, at once deduce the addition-theorem for g) (u + v).
Evidently

which is only one of many forms : one of the most useful is

which can be deduced from the preceding form.

The result can be used to modify the expression for a general doubly-periodic function
* (z) obtained in § 128. We have

Each derivative of f can be expressed either as an integral algebraical function of $ (z - a,.)
or as the product of jjp' (z — ar) by such a function ; and by the use of the addition-theorem
these can be expressed in the form

L >
where L, M, N are rational integral algebraical functions of $(z). Hence the function

can be expressed in the same form, the simplest case being when all its infinities are
simple, and then

4. (z) = C+ 2 Ar{(e-ar)

(*)-§» (Or)

with the condition 2 Ar = 0.

r=l

Ex. 5. The function $ (z) — e1 is an even function, doubly-periodic in 2« and 2o> and
having 2 = 0 for an infinity of the second degree ; it has only a single infinity of the second
degi'ee in a fundamental parallelogram.

Again, z = &> is a zero of the function ; and, since ^X («) = 0 but $>" (o>) is not zero, it is a
double zero of $ (z)-el. All the zeros are therefore reducible to 2 = o> ; and the function
has only a single zero of the second degree in a fundamental parallelogram.

Taking then the parallelogram of reference so as to include the points 0 = 0 and 0=o>,
we have

where Q (z) has no zero and no infinity for points within the parallelogram.

Again, for g> (z + o>) - e± , the irreducible zero of the second degree within the parallelo-

131.] OF DOUBLY-PERIODIC FUNCTIONS 263

gram is given by S + <B = O>, that is, it is 0 = 0; and the irreducible infinity of the second
degree within the parallelogram is given by z + a = 0, that is, it is z = v. Hence we have

where Ql (z) has no zero and no infinity for points within the parallelogram.

Hence {£> (z} - ej {%> (z + «) - ej m Q (z) Q1 (z\

that is, it is a function which has no zero and no infinity for points within the
parallelogram of reference. Being doubly-periodic, it therefore has no zero and no infinity
anywhere in the plane ; it consequently is a constant, which is the value for any point.
Taking the special value s = a>, we have jp(m') = es, and (jf>(a>' + a>) = e2 ; and therefore

{#> (*) ~ e,} (V (* + «)- e,} = (e3 - *i) (e, - *i).

Similarly {#> (z) - e2} {#> (z + »") - e2} = (ex - e2) (es - e2),

and {#> (2) - ^ {p (z + a)') - e3} = (e2 - e3} (^ - <?3).

It is possible to derive at once from these equations the values of the ^-function for
the quarter-periods.

Note. In the preceding chapter some theorems were given which indicated that
functions, which are doubly-periodic in the same periods, can be expressed in terms of one
another : in particular cases, care has occasionally to be exercised to be certain that the
periods of the functions are the same, especially when transformations of the variables are
effected. For instance, since g) (z) has the origin for an infinity and sn u has it for a zero,
it is natural to express the one in terms of the other. Now $ (z) is an even function, and
sn u is an odd function ; hence the relation to be obtained will be expected to be one
between ®(z) and sn2w. But one of the periods of sn2 u is only one-half of the correspond
ing period of sn u ; and so the period-parallelogram is changed. The actual relation* is

(P (z) - <?3 = (<?!- e3) sn-2«,
where u = (el -esf z and F = (<?2 -e^)l(el -e:i).

Again, with the ordinary notation of Jacobian elliptic functions, the periods of sn z are
4 A" and 2iA", those of dn z are 2 A and 4i K', and those of en z are 4 A' and 2A'+ 2iA''. The
squares of these three functions are homoperiodic in 2K and ZiK' ; they are each of the
second order, and they have the same infinities. Hence sn2 z, en2 z, dn2 z are equivalent to
one another (§ 116, V.).

But such cases belong to the detailed development of the theory of particular classes of
functions, rather than to what are merely illustrations of the general propositions.

132. As a last illustration giving properties of the functions just
considered, the derivatives of an elliptic function with regard to the periods
will be obtained.

Let (/> (z) be any function, doubly-periodic in 2o> and 2&/ so that

</> (z + 2m&> + 2m V) = </> (z),

the coefficients in <f> implicitly involve &> and CD'. Let <f>1} <£2, and </>' respec
tively denote 90/3t», 9<£/9o/, 9^/9^ ; then

^ (z + 2771&) + 2m' to') + %m<j>' (z + 2mo> + 2wV) = fa (z),
fa (z + 2m&) + 2??iV) -1- 2m'<£' (z + 2rao> + 2wV) = fa (z),

$ (z + 2m&) + 2m V) = <f> (z).

" Halphen, Fonctions Elliptiques, t. i, pp. 23 — 25.

264 PERIOD-DERIVATIVES [132.

Multiplying by &>, ro', z respectively and adding, we have
&></>! (z + 2mo> + 2m V) + o>'</>2 (z + 2m<« + 2mV)

+ (z + 2mw + 2wV) </>' 0 + 2mo)

= 0)0! (Z) + 0)'(f).2 (Z) + Z<f>' (Z).

Hence, if f(z} = mfa (z) + 6/02 (z) + z$ (z),

then f(z) is a function doubly -periodic in the periods of (f>.

Again, multiplying by rj, 77', %(z), adding, and remembering that
£0 + 2mm + 2raV) = £($)

we have

77$! (z + 2mw + 2m V) + 7/<£2 0 + 2m«o + 2m V)

+ £(z + 2mm + 2m' ay') <f>' (z + 2mw + 2m'a>')

-ik<fi)+J+,(*) + f(*Wto

Hence, if g (z) = yfa (z) + q'fa (z) + £ (z) $' (z),

then g(z) is a function doubly-periodic in the periods of <f>.

In what precedes, the function <f>(z) is any function, doubly-periodic in
2o>, 2&)' ; one simple and useful case occurs when 0 (z) is taken to be the
function z. Now

and fW-J-^

hence, in the vicinity of the origin, we have

9P >d@ d& 2

(o ^- + 03 5*7 + jp-«- as — + even integral powers of z1
d(o d(o dz z-

= -2^>,

since both functions are doubly-periodic and the terms independent of z
vanish for both functions. It is easy to see that this equation merely
expresses the fact that <p, which is equal to

l

is homogeneous of degree — 2 in z, &>, to'.
Similarly

9|J> / ty ^d%> 22

77 2+*) ~r-/ + b (-2r) y- = - ~i + j^ 9-i + even integral powers of z.

But, in the vicinity of the origin,

,5-7 = — + YQ^ -I- even integral powers of 2,

132.] OF WEIEHSTRASS'S FUNCTION 265

so that

9P / d@ »/ \ dP 1 32lP • x r

17 3^- + V g :S + f<«) |^t i gji ™i** even mtegral powers of z.

The function on the left-hand side is doubly-periodic : it has no infinity
at the origin and therefore none in the fundamental parallelogram ; it there
fore has no infinities in the plane. It is thus constant and equal to its value
anywhere, say at the origin. This value is ^gz> and therefore

T/w's equation, when combined with

,

+ eo' ; + z = - ty,
dco da> oz

j

gives the value of ~- and -^ , .
J dm 9&)

The equations are identically satisfied. Equating the coefficients of z2 in
the expansions, which are valid in the vicinity of the origin, we have

and equating the coefficients of ^ in the same expansions, we have

Hence for any function u, which involves w and &/ and therefore implicitly
involves g2 and ^r3, we have

du ,du
w 5- + w — , =

aw a&)

9w . , 3w
17 a- +T;' — = -
9&) 9&)

Since ^) is such a function, we have

f : *'

being ^/te equations which determine the derivatives of $ with regard to the
invariants g., and g.^.

266 EVEN [132.

The latter equation, integrated twice, leads to
9V da- 2 80- 1

a differential equation satisfied by <r(z)*.

133. The foregoing investigations give some of the properties of doubly-
periodic functions of the second order, whether they be uneven and have two
simple irreducible infinities, or even and have one double irreducible infinity.

If a function U of the second order have a repeated infinity at z = y, then
it is determined by an equation of the form

or, taking U - £ (X + fi + v) = Q, the equation is

Q'» = 4a2 [(Q - e,) (Q - e,) (Q - $,)]*,

where ^ + e2 + e3 = 0. Taking account of the infinities, we have

Q=@(az- ay) ;

and therefore U-±(\ + /Ji. + v) = %> (az - ay)

. . 1 (tp'(az) + cp (ay)}'2
= -Q (az) - <o (ay) + — —

a x ' o x "

by Ex. 4, p. 262. The right-hand side cannot be an odd function; hence
an odd function of the second order cannot have a repeated infinity. Similarly,
by taking reciprocals of the functions, it follows that an odd function of the
second order cannot have a repeated zero.

It thus appears that the investigations in §§ 120, 121 are sufficient for the
included range of properties of odd functions. We now proceed to obtain
the general equations of even functions. Every such function can (by § 118,
XIII., Cor. I.) be expressed in the form |a#> (z) +b}+ {c#> (z) + d], and its
equations could thence be deduced from those of p(z)\ but, partly for
uniformity, we shall adopt the same method as in § 120 for odd functions.
And, as already stated (p. 251), the separate class of functions of the second
order that are neither even nor odd, will not be discussed.

134. Let, then, <j>(z) denote an even doubly-periodic function of the
second order (it may be either of the first class or of the second class) and let
2&), 2<w' be its periods ; and denote 2&) + 2ft)' by 2o>". Then

since the function is even ; and since

<£ (ft) + Z) = <f> (— &) — z)

= <f> (2&) — &) — z)
= (j) (CD — *),

* For this and other deductions from these equations, see Frobenius und Stickelberger, Crelle,
t. xcii, (1882), pp. 311—327; Halphen, Traite des feme t ions elliptiqucs, t. i, (1886), chap. ix. ;
and a memoir by the author, quoted on p. 254, note.

134.] DOUBLY-PERIODIC FUNCTIONS

it follows that <£ (&> + z)— and, similarly, $ (&>' + z) and 0 (to" + z) are even
functions.

Now </> (w + a), an even function, has two irreducible infinities, and is
periodic in 2&>, 2&/ ; also <£ (z), an even function, has two irreducible infinities
and is periodic in 2&>, 2&/. There is therefore a relation between 0 (z) and
</> (w +z), which, by § 118, Prop. XIII., Cor. I., is of the first degree in <£ (z) and
of the first degree in <j) (&> + z) ; thus it must be included in

B<f> (z) <j>(<o + z)-C<l> (z) -C'<t>(a> + z)+A = 0.

But <£ (z) is periodic in 2<w ; hence, on writing z + <w for z in the equation, it

becomes

B<f>(a>+z)<j>(z)-C<f>(co+z)-C'<l>(z) + A=0;

thus tf=C".

If B be zero, then (7 may not be zero, for the relation cannot become
evanescent : it is of the form

A' .............................. (1).

If B be not zero, then the relation is

Treating <f> (w + z) in the same way, we find that the relation between it

and (f) (z) is

F(j> (z) (f> (ay' + z)-D(j> (z} -D(j>(a>' + z) + E = 0,

so that, if F be zero, the relation is of the form

£(*) + 0(a>' + *) = J0' ........................... (I)',

and, if F be not zero, the relation is of the form

Four cases thus arise, viz., the coexistence of (1) with (1)', of (1) with (2)',
of (2) with (1)', and of (2) with (2)'. These will be taken in order.

I. : the coexistence of (1) with (1)'. From (1) we have

<j> (a> + z} + (j> (&>" + z) = A',

so that </> (z) + <f) (w + z) + (f) (w + z) + 0 (<w" + z) = 2A'.

Similarly, from (1)',

so that A = E', arid then

(f)((0 + z)=(j)((o'+ Z\

whence <w ~ &>' is a period, contrary to the initial hypothesis that 2&> and 2&>'
determine a fundamental parallelogram. Hence equations (1) and (1)' cannot
coexist.

268 EVEN [134.

II. : the coexistence of (1) with (2)'. From (1) we have
<£(«" + z} = A' - <£(&>' + z)

on substitution from (2)'. From (2)' we have

cb (co + z) = -5*1) -- ( — =r
F<f) (CD + z) - D

_ (A'D -E)-D<j> (z)
= A'F - D - F(f> (z) '
on substitution from (1). The two values of <£ (&>" + z) must be the same,

whence

A'F-D = D,

which relation establishes the periodicity of </> (z) in 2ft)", when it is considered
as given by either of the two expressions which have been obtained. We

thus have

A'F=W-
and then, by (1), we have

<f>(z)-j+<l>(
and, by (2)', we have

If a new even function be introduced, doubly-periodic in the same periods
having the same infinities and defined by the equation

& 0) = </> 0) - J >
the equations satisfied by fa (z) are

fa(a> + z) + fa(z) = 0 }

fa (&)' + z) fa (z) = constant] '

To the detailed properties of such functions we shall return later ; meanwhile
it may be noticed that these equations are, in form, the same as those satisfied
by an odd function of the second order.

III. : the coexistence of (2) with (1)'. This case is similar to II., with the
result that, if an even function be introduced, doubly-periodic in the same
periods having the same infinities and defined by the equation

C

fa (Z) = <£ (2) - -g ,

the equations satisfied by fa (z) are

fa (&>' + z) + fa (z) = 0 }

fa (&) + z) fa (z) = constant] '

It is, in fact, merely the previous case with the periods interchanged.

134.] DOUBLY-PERIODIC FUNCTIONS 269

IV. : the coexistence of (2) with (2)'. From (2) we have

_ (CD - AF) <ft (z) - (GE - AD)
~ (BD - CF) <J> (z) - (BE - CD) '

on substitution from (2)'. Similarly from (2)', after substitution from (2), we
have

~

The two values must be the same ; hence

CD-AF=-(GD-BE\

which indeed is the condition that each of the expressions for <ft (&>" + z)
should give a function periodic in 2&>". Thus

One case may be at once considered and removed, viz. if C and D vanish
together. Then since, by the hypothesis of the existence of (2) and of (2)',
neither B nor F vanishes, we have

A__E

B~ F'

so that u + , =

and then the relations are <£ (&> + z) + <f) (&>' + z) = 0,

or, what is the same thing, <ft (Y) + <ft (&>" + z) — 0 ]

and </> (z) </>(&> + z) = constant j '

This case is substantially the same as that of II. and III., arising merely

from a modification (§ 109) of the fundamental parallelogram, into one whose

sides are determined by 2&> and 2&>".

Hence we may have (2) coexistent with (2)' provided

AF + BE=WD;
C and D do not both vanish, and neither B nor F vanishes.

IV. (1). Let neither C nor D vanish ; and for brevity write

<f>((o + z)=<l>1, (f> (w" + z) = <£o, </> (&)' + z) = $3, (f) (z) = 0.
Then the equations in IV. are

Now a doubly-periodic function, with given zeros and given infinities, is
determinate save as to an arbitrary constant factor. We therefore introduce
an arbitrary factor X, so that

<£=Xi/r,

G D

and then taking = CI' = Ca'

270 EVEN [134.

£
we have (^ - Cl) (fa - Cj) = d2- -^ ,

ET

The arbitrary quantity A, is at our disposal : we introduce a new quantity c2,
defined by the equation

A

Tt-. o — Ci (C2 + €3) C2C3 ,

and therefore at our disposal. But since

AF + BE=2CD,

A E .CD

we have ^ + ^ = 2 ^ ^ = 2Clc3j

ri

and therefore ^--2 = c3 (Cj + c2) - 0^2 .

Hence the foregoing equations are

- d) = (Cj - C2) (d - C3),

- C3) = (C3 - d) (C3 - C2).

The equation for ^>2, that is <f>((o" + z), is

_Lcf)-M

where L= CD - BE = AF - CD, M=AD-CE, N=CF-BD,

so that ^ + 5M" = 2CL.

As before, one particular case may be considered and removed. If N be

zero, so that

C_D_

B~F~a

AE CD ,.

say, and B+F=RF= '

then we find $ + ^>2 = ^>i + <£3 = 2«,

or taking a function ^ = 0 — a,

the equation becomes % (^ + % C^" + ^ = 0.
The other equations then become

and therefore they are similar to those in Cases II. and III.
If N be not zero, then it is easy to shew that
N=BF\(c1-c3)>

M = BF\3 (d - c3) (c.,C! + c,c3 - dc3) ;

134.] DOUBLY-PERIODIC FUNCTIONS 271

and then the equation connecting 0 and 02 changes to

s - Ca) = (Ca - Cx) (Ca - Cs)

which, with (^ — d) ("^i — d) = (d — c2) (d — c3)

( r — ^3/ \ i 3 — ^3/ == V^3 ^" ^3 ^2'

are relations between ty, ^rl} -^2, ty.3, where the quantity c2 is at our disposal.

IV. (2). These equations have been obtained on the supposition that
neither G nor D is zero. If either vanish, let it be C: then D docs not
vanish ; and the equations can be expressed in the form

E

D\

J

E\ E(D*-EF)

We therefore obtain the following theorem :

If (f> be an even function doubly-periodic in 2&> and 2&>' and of the second
order, and if all functions equivalent to <J> in the form R<f> + 8 (where R and
S are constants) be regarded as the same as 0, then either the function satisfies
the system of equations

00) 0O"
where H is a constant ; or it satisfies the system of equations

{0 0) - d} {0 (ft) +Z)- d] = (Ci - C2) (d - C3)
{00)-C3}{0(>/ +^)-C3}=(Cs-C1)(Cs-Ca)
{0 0) - C2} (0 ((,)" + Z)- C2} = (C2 ~ d) (C2 - Cs)

where of the three constants clt c2, cs one can be arbitrarily assigned.
We shall now very briefly consider these in turn.

135. So far as concerns the former class of equations satisfied by an even
doubly-periodic function, viz.,

we proceed initially as in (§ 120) the case of an odd function. We have the
further equations

00) = 0(-4

0 (ft) + Z) — 0 (ft) — Z), 0 (a/ + Z) = 0 (ft)' — Z).

* The systems obtained by the interchange of w, w', w" among one another in the equations
are not substantially distinct from the form adopted for the system I. ; the apparent difference
can be removed by an appropriate corresponding interchange of the periods.

272 EVEN DOUBLY-PERIODIC FUNCTIONS [135.

Taking z = — ^w, the first gives

so that ^&> is either a zero or an infinity.
If \<£> be a zero, then

(f> (f to) = $ (<« + ^ft)) = — <f> (^») by the first equation

= 0,
so that ^&> and f&> are zeros. And then, by the second equation,

&)' + ^<w, &)' 4- f a)
are infinities.

If \w be an infinity, then in the same way |w is also an infinity ; and
then a)' + \w, &>' + f &) are zeros. Since these amount merely to interchanging
zeros and infinities, which is the same functionally as taking the reciprocal of
the function, we may choose either arrangement. We shall take that which
gives ^0), f &> as the zeros ; and &>' 4- ^&>, &/ + f &> as the infinities.

The function <j> is evidently of the second class, in that it has two distinct
simple irreducible infinities.

Because &>' + |&), &>' + f &> are the irreducible infinities of </> (z), the four
zeros of $' (z} are, by § 117, the irreducible points homologous with &>",
&)" + &>, &>" + a)', a)" + &)", that is, the irreducible zeros of (f)' (z) are 0, &>, &>', &>".
Moreover

by the first of the equations of the system ; hence the relation between (f> (
and ((>' (z) is

#* (z} = A{<t>(z)-$ (())} {</> (z) - (/> («)} |0 (*) - (/> (ft)')} {(/> (*) - </> («")}

= A [p (0) - p (z)}{p (ft)') - ^ (*)}.
Since the origin is neither a zero nor an infinity of <£ (^), let

so that </>j (0) is unity and 0/ (0) is zero ; then

^(*)«X»{l-^(*)}{^-^(f))

the differential equation determining fa (z).

The character of the function depends upon the value of p and the
constant of integration. The function may be compared with en u, by taking

2ft), 2&/ = 4>K, 2K + 2iK' ; and with — *— , by taking 2ft), 2ft)' = 2K, MK',

dn u

which (§ 131, note) are the periods of these (even) Jacobian elliptic functions.
We may deal even more briefly with the even, function characterised by
the second class of equations in § 134. One of the quantities c1} c2, c3 being
at our disposal, we choose it so that

Ci + c2 + c3 = 0 ;

and then the analogy with the equations of Weierstrass's ^-function is
complete (see § 133).

CHAPTER XII.

PSEUDO-PERIODIC FUNCTIONS.

136. MOST of the functions in the last two Chapters are of the type
called doubly-periodic, that is, they are reproduced when their arguments are
increased by integral multiples of two distinct periods. But, in §§ 127, 130,
functions of only a pseudo-periodic type have arisen : thus the ^-function
satisfies the equation

m2&> + m'2&>') = £(» + m2i) + m'2v',
,nd the cr-function the equation

m' i (mr,+m'r,') (z+wuo+m'oi1)

These are instances of the most important classes: and the distinction
between the two can be made even less by considering the function
e^(z} — ^(z), when we have

£ (z + ra2&> + m'2&>') = e-mr> e"™'*' % (z).

In the case of the ^-function an increase of the argument by a period leads
to the reproduction of the function multiplied by an exponential factor that
is constant, and in the case of the <r-function a similar change of the
argument leads to the reproduction of the function multiplied by an
exponential factor having its index of the form az + b.

Hence, when an argument is subject to periodic increase, there are three
simple classes of functions of that argument.

First, if a function f(z) satisfy the equations

/(* + 2fi>) =/(*), /(* + 2«') =/(*),

it is strictly periodic : it is sometimes called a doubly-periodic function of the
first kind. The general properties of such functions have already been
considered.

Secondly, if a function F(z) satisfy the equations

F (z + 2&>) = pF (z), F (z + 2&/) - pfF (z),
F- 18

274 THREE KINDS [136.

where /u, and fjf are constants, it is pseudo-periodic : it is called a doubly-
periodic function of the second kind. The first derivative of the logarithm
of such a function is a doubly-periodic function of the first kind.

Thirdly, if a function <f> (z) satisfy the equations

<j>(z + 2o)) = eaz+b <j> (z\ <f>(z + 2ft)') = ea'z+v (j> (z),

where a, b, a', b' are constants, it is pseudo-periodic : it is called a doubly-
periodic function of the third kind. The second derivative of the logarithm
of such a function is a doubly-periodic function of the first kind.

The equations of definition for functions of the third kind can be
modified. We have

. <f> (Z + 2ft) + 2ft)') = e«(2+2<o')+6+a'z+6' <£ (z)
— ga' (2+2o>) +b'+az+b J, fz\

whence a'oo — am' = — nnri,

where ra is an integer. Let a new function E (z) be introduced, defined by

the equation

£(«)«*"+*• t(*)i

then X and /A can be chosen so that E (z} satisfies the equations

E(z + 2a)) = E (z\ E(z+ 2ft)') = eAz+£ E (z\
From the last equations, we have

E (z + 2&) + 2ft)') = eA(*+**+B E ^

= eAz+s E (z),
so that 2Aa) is an integral multiple of 2?™'.

Also we have E(z + 2o>) = e*(*-*»'+^+a») <j>(z + 2o>)

so that 4X&) + a = 0,

and 4A,ftr + 2/A&) +6 = 0 (mod.

Similarly, E (z + 2ft)') = e^+wj'+^+w, 0 ^ + 2ft)')

so that 4Xo)' + a = A,

and 4W2 + 2/^co' + 6' = B (mod. 27ri).

From the two equations, which involve X and not //,, we have

Aco = a'o) — aw'

agreeing with the result with 2 A co is an integral multiple of Ziri.

And from the two equations, which involve /j,, we have, on the elimination
of /j, and on substitution for X and A,

b'co — 6ft)' — a&)' (ft)' — &)) = 5ft) (mod. 2-Tn').

136.] OF DOUBLY-PERIODIC FUNCTIONS 275

If A be zero, then E(z) is a doubly-periodic function of the first kind
when eB is unity, and it is a doubly-periodic function of the second kind
when eB is not unity. Hence A, and therefore m, may be assumed to be
different from zero for functions of the third kind. Take a new function
3?z such that

mm
then <l> (z) satisfies the equations

4) (z + 2&)) = <I> (z\ <&(z + 2o)') = e w 3>(z)

* / \ /' \ / \ /t

which will be taken as the canonical equations defining a doubly -periodic
function of the third kind.

Ex. Obtain the values of X, p, A, B for the Weierstrassian function ir(z).

We proceed to obtain some properties of these two classes of functions
which, for brevity, will be called secondary-periodic functions and tertiary-
periodic functions respectively.

Doubly-Periodic Functions of the Second Kind.

For the secondary-periodic functions the chief sources of information are

Hermite, Comptes Rendus, t. liii, (1861), pp. 214—228, ib., t. Iv, (1862), pp. 11—18,
85 — 91 ; Sur quelques applications des fonctions elliptiques, §§ I — in, separate
reprint (1885) from Comptes Rendus; "Note sur la theorie des fonctions ellip
tiques" in Lacroix, vol. ii, (6th edition, 1885), pp. 484—491; Cours d' Analyse,
(4me ed.), pp. 227—234.

Mittag-Leffler, Comptes Rendus, t. xc, (1880), pp. 177 — 180.

.Frobenius, Crelle, t. xciii, (1882), pp. 53 — 68.

Brioschi, Comptes Rendus, t. xcii, (1881), pp. 325—328.

Halphen, Traite' des fonctions elliptiques, t. i, pp. 225 — 238, 411 — 426, 438 442, 463.

137. In the case of the periodic functions of the first kind it was proved
that they can be expressed by means of functions of the second order in the
same period — these being the simplest of such functions. It will now be
proved that a similar result holds for secondary- periodic functions, defined by
the equations

Take a function Q (z} =

a (z) a- (a)

then we have G(z+2a>) = <r(* + g

a (a) a (z + 2w)

arid G(z+ 2&/) = e'V«+2W Q. (^).

The quantities a and X being unrestricted, we choose them so that

„ _ g2rja+2A<o ' __ g2T)'a+2A(o' •

and then G (z), a known function, satisfies the same equation as F (z).

18—2

276 PSEUDO-PERIODIC FUNCTIONS [137.

Let u denote a quantity independent of z, and consider the function

f(Z) = F(z)G(u-z}.
We have f(z + 2o>) = F(z + 2o>) G (u- z - 2w)

=/(*) ;

and similarly f(z + 2<o') =f(z),

so that/(X) is a doubly-periodic function of the first kind with 2« and 2o>'

for its periods.

The sum of the residues of f(z) is therefore zero. To express this sum,
we must obtain the fractional part of the function for expansion in the
vicinity of each of the (accidental) singularities of f(z), that lie within the
parallelogram of periods. The singularities of/ (2) are those of G (u — z) and
those of F(z).

Choosing the parallelogram of reference so that it may contain u, we have
z = u as the only singularity of G (u — z) and it is of the first order, so that,
since

$(£) — =+ positive integral powers of f
in the vicinity of £= 0, we have, in the vicinity of u,
f(z) = {F (u) + positive integral powers of u — z} \ — -4- positive powers I

= -- — + positive integral powers of z — u ;

hence the residue of/(Y) for u is —F(u}.

Let z = c be a pole of F (z) in the parallelogram of order n + 1 ; and, in
the vicinity of c, let

(?! _ cf / 1 \ „ dn ( 1 \

F(z) = ^—c +G^Z (jr^J + • • • + C'n+i fan (zITc) + P°sltlve integral powers.

Then in that vicinity

and therefore the coefficient of - in the expansion of f(z) for points in the

Z ~~ 0

vicinity of c is

which is therefore the residue off(z) for c.

This being the form of the residue of f(z) for each of the poles of F (z),
then, since the sum of the residues is zero, we have

137.] OF THE SECOND KIND 277

or, changing the variable,

,. ..n+l n -

where the summation extends over all the poles of F(z) within that parallelo
gram of periods in which z lies. This result is due to Hermite.

138. It has been assumed that a and \, parameters in 0, are determinate,
an assumption that requires /j, and ^ to be general constants : their values
are given by

yd 4- &>X = | log fjb, r)'a + &/X — \ log //,

and, therefore, since ijca' — rfca = ± ^ITT, we have

+ ITTCL = w' log /JL — co log //)

+ iir\ = — V) log /i + 77 log /z'j '

Now X may vanish without rendering G (z) a null function. If a vanish (or,
what is the same thing, be an integral combination of the periods), then G (z)
is an exponential function multiplied by an infinite constant when X does not
vanish, and it ceases to be a function when X does vanish. These cases must
be taken separately.

First, let a and X vanish* ; then both //, and /// are unity, the function F
is doubly-periodic of the first kind ; but the expression for j^is not determinate,
owing to the form of G. To render it determinate, consider X as zero and a
as infinitesimal, to be made zero ultimately. Then

„,, o-(z) + aa'(z) + ... .^

(*(z) = - - ~ — — (1 + positive integral powers of a)

= - + £ (z) + positive powers of a.
a

Since a is infinitesimal, /JL and /j,' are very nearly unity. When the
function F is given, the coefficients C1} <72, ... may be affected by a, so that
for any one we have

Ck — bk + ayk + higher powers of a,

where yh is finite ; and bk is the actual value for the function which is strictly
of the first kind, so that

Sk-O,

the summation being extended over the poles of the function. Then retaining
only a"1 and a°, we have

This case is discussed by Hermite (I.e., p. 275).

278 MITTAG-LEFFLER'S THEOREM [138.

where C0, equal to £71, is a constant and the term in - vanishes. This expres-

CL

sion, with the condition S^ = 0, is the value of F (u) or, changing the variables,
we have

with the condition S&i = 0, a result agreeing with the one formerly (§ 128)
obtained.

When F is not given, but only its infinities are assigned arbitrarily, then
SO = 0 because F is to be a doubly-periodic function of the first kind ; the

term - "£C vanishes, and we have the same expression for F(z) as before.
Secondly, let a vanish* but not \, so that ^ and // have the forms

We take a function g (z) =

then g(z- 2o>) = ^ e^ £ (z - 2eo )

and g(z-2a>') = p'-1 {g (z} - 2?/ e^} .

Introducing a new function H (z) defined by the equation

we have H (z + 2t») = H (z) - 2ijeA <«-*» F (z),

and H (z + 2o>') = H (z) - 27?V<M-*> F(z).

Consider a parallelogram of periods which contains the point u ; then, if © be

the sum of the residues of H (z) for poles in this parallelogram, we have

the integral being taken positively round the parallelogram. But, by § 116,
Prop. II. Cor., this integral is

f e-*(p+*-«) F (p + 2tot) dt - 0/77 f e-^+
Jo Jo

where p is the corner of the parallelogram and each integral is taken for real
values of t from 0 to 1. Each of the integrals is a constant, so far as concerns
u ; and therefore we may take

® = -Ae^u,

the quantity inside the above bracket being denoted by —\i-rrA.

The residue of H (z) for z = u, arising from the simple pole of g (u — z), is
-F(u) as in§ 137.

If z = c be an accidental singularity of F (z) of order n+1, so that, in the
vicinity of z = c,

F(.) = C, + 0. A- + . . . + BU i- + P (, - c),

This is discussed by Mittag-Leffler, (I.e., p. 275).

138.] ON SECONDARY FUNCTIONS 279

then the residue of H (z) for z = c is

d dn

and similarly for all the other accidental singularities of F (z}. Hence

F(z) = A** +

where the summation extends over all the accidental singularities of F (z) in a
parallelogram of periods which contains z, and y (z) is the function exz%(z}.
This result is due to Mittag-Leffler.

Since /* = e2*" and

g (z - c + 2&>) = fig (z - c) +
we have

and therefore 2 (Gl + C.2\ 4- . . . + Gn+l\n) e~^ = 0,

the summation extending over all the accidental singularities of F(z). The

same equation can be derived through ^F(z) = F(z + 2&>').

Again 2(7: is the sum of the residues in a parallelogram of periods, and
therefore

the integral being taken positively round it. If p be one corner, the integral

n

F (p + 2co't) dt,

Jo

IS

/•i n

o
, each integral being for real variables of t.

Hermite's special form can be derived from Mittag-Leffler's by making \
vanish.

Note. Both Hermite and Mittag-Leffler, in their investigations, have
used the notation of the Jacobian theory of elliptic functions, instead of
dealing with general periodic functions. The forms of their results are as
follows, using as far as possible the notation of the preceding articles.

I. When the function is denned by the equations

F (z + 2K) = ^F (z), F(z+ 2iK') = ^F (z),

then F(z) =

280 INFINITIES AND ZEROS [138.

(the symbol H denoting the Jacobian .ff-function), and the constants <w and X
are determined by the equations

II. If both X and to be zero, so that F(z) is a doubly-periodic function
of the first kind, then

with the condition 5$i = 0.

III. If W be zero, but not X, then

...
where g (z} = --& V,

the constants being subject to the condition

2 (G, + C,\ + . . . + Gn+1 X")e-Ac = 0,

and the summations extending to all the accidental singularities of F(z) in a
parallelogram of periods containing the variable z.

139. Reverting now to the function F(z) we have G (z), defined as

a (z) a (a)

when a and X are properly determined, satisfying the equations
G(z + 2a>) = ftG (z), £0 + 2&/) = yu/£0).

Hence H (z) = F(z)/G (z) is a doubly-periodic function of the first kind ; and
therefore the number of its irreducible zeros is equal to the number of its
irreducible infinities, and their sums (proper account being taken of multipli
city) are congruent to one another with moduli 2« and 2&>'.

Let Ci, c2,..., cm be the set of infinities of F (z) in the parallelogram of
periods containing the point z ; and let y:, . . . , 7^ be the set of zeros of F (z) in
the same parallelogram, an infinity of order n or a zero of order n occurring
n times in the respective sets. The only zero of 0 (z) in the parallelogram is
congruent with — a, and its only infinity is congruent with 0, each being
simple. Hence the m+l irreducible infinities of H (z) are congruent with

a, GI, GZ, . . . , cm,
and its /* + 1 irreducible zeros are congruent with

0, 71, 7s> •••>%*;
and therefore m + 1 = p, + 1,

139.] OF SECONDARY FUNCTIONS 281

From the first it follows* that the number of infinities of a doubly-periodic
function of the second kind in a parallelogram of periods is equal to the number
of its zeros, and that the excess of the sum of the former over the sum of the
latter is congruent with

, (°>' i w i ,
+ — , log it -. log u,

- \7Tl TTl 6 '

/

the sign being the same as that of 9t ( —

\10)

The result just obtained renders it possible to derive another expression
for F (z), substantially due to Hermite. Consider a function

F (Z) = °-Q-7i) 0-0-72).. -0-0-7™) ePZ

(T(z-c1)(r(z-c2)...ar(z-cm) '

where p is a constant. Evidently F1 (z) has the same zeros and the same
infinities, each in the same degree, as F (z). Moreover

F, (Z + 2ft)) = Fl (Z) e2,(2C-2y) + 2pWj
F1 (Z + 2ft)') = F! (Z) e2V(2e-2y)+2P«,'t

If, then, we choose points c and 7, such that

Sc — £7 = a,
and we take p = \ where a and X are the constants of G (z), then

F, (z + 2co) = ^ (z), F, (z + 2ft>') = n'Fj. (z).

The function Fl (z)/F(z) is a doubly-periodic function of the first kind and by
the construction of Fl (z) it has no zeros and no infinities in the finite part of
the plane: it is therefore a constant. Hence

F(z] = A gfr-'ftM*- •/»)•••* (*—*») ^

a(z- c,) a- (z - C2). . .o- (z - Cm)

where Sc — £7 = a, and a and A, are determined as for the function G (z}.

140. One of the most important applications of secondary doubly-periodic
functions is that which leads to the solution of Lame's equation in the cases
when it can be integrated by means of uniform functions. This equation is
subsidiary to the solution of the general equation, characteristic of the
potential of an attracting mass at a point in free space; and it can be
expressed either in the form

jY = (Ak'2 sn2 z + B) w,
or in the form - 2 - = (A@ (z) + B} w,

* Frobenius, Crelle, xciii, pp. 55 — 68, a memoir which contains developments of the properties
of the function G (z). The result appears to have been noticed first by Brioschi, (Comptes Ilendus,
t. xcii, p. 325), in discussing a more limited form.

282 LAMP'S [140.

according to the class of elliptic functions used. In order that the integral
may be uniform, the constant A must be n (n -f 1), where n is a positive
integer ; this value of A, moreover, is the value that occurs most naturally in
the derivation of the equation. The constant B can be taken arbitrarily.

The foregoing equation is one of a class, the properties of which have
been established* by Picard, Floquet, and others. Without entering into
their discussion, the following will suffice to connect them with the secondary
periodic function.

Let two independent special solutions be g (z) and h (z), uniform functions
of z ; every solution is of the form ag (z} + /3h (z}, where a and /3 are constants.
The equation is unaltered when z + 2w is substituted for z ; hence g {z + 2&>)
and h (z + 2&>) are solutions, so that we must have

g (z + 2w) = Ag (z} + Bh (z}, h(z + 2o>) = Cg (z) + Dh (z\

where, as the functions are determinate, A, B, C, D are determinate constants,
such that AD — BC is different from zero.

Similarly, we obtain equations of the form

g (z + 2co') = A'g (z) + B'h (z\ h(z + 2co') = C'g (z} + D'h (z}.
Using both equations to obtain g (z + 2o> + 2&/) in the same form, we have

BC' = B'C, AB' + BD' = A'B + B'D ;
and similarly, for h (z + 2w + 20)'), we have

C G' A-D A'-U

therefore -~ = -™ = o, — ~ — = — ™ — = e.

x> -D n n

Let a solution F (z} = ag (z) + bh (z)

be chosen, so as to give

if possible. The conditions for the first are

a b

so that a/b (= £) must satisfy the equation

and the conditions for the second are

aA' + bCf aB' + bD'

* Picard, Comptes Rendus, t. xc, (1880), pp. 128—131, 293—295; Crelle, t. xc, (1880), pp.
281—302.

Floquet, Comptes Rendus, t. xcviii, (1884), pp. 82 — 85 ; Ann. de VEc. Norm. Sup., 3mc Ser.,
t. i, (1884), pp. 181—238.

140.] DIFFERENTIAL EQUATION 283

so that £ must satisfy the equation

A'-D'=^B'~~.
These two equations are the same, being

p.-«g-ft*a

Let £j and £2 be the roots of this equation which, in general, are unequal ; and
let fa, fa and fa, fa.' be the corresponding values of /z, //. Then two functions,
say FI (z) and F^ (z), are determined : they are independent of one another, so
therefore are g (z) and h (z) ; and therefore every solution can be expressed in
terms of them. Hence a linear differential equation of the second order, having
coefficients that are doubly-periodic functions of the first kind, can generally be
integrated by means of doubly -periodic functions of the second kind.

It therefore follows that Lame's equation, which will be taken in the form

can be integrated by means of secondary doubly-periodic functions.

141. Let z = c be an accidental singularity of w of order m ; then, for
points z in the immediate vicinity of c, we have

and therefore

2mp

~ z- c + P°SltlVe P°wers °f * -

Since this is equal to n (n + 1) @ (z) + B

it follows that c must be congruent to zero and that m, a positive integer,
must be n. Moreover, p = 0. Hence the accidental singularities of w are
congruent to zero, and each is of order n.

The secondary periodic function, which has no accidental singularities
except those of order n congruent to z = 0, has n irreducible zeros. Let them
be — alt — a2,..., — an; then the form of the function is

Hence 1 *? = ,-»?« +

or, taking p = - ^(ar), we have

and therefore i *? - 1 (*?)' . n(> (,) - X f> (« + «,
19 O^ W2\dzj * v y r»i

284 INTEGRATION

But, by Ex. 3, § 131, we have

[141.

4 r=1 £> (ar) - p (z)

,

by Ex. 4, § 131. Thus

W

Now

r=l S=l

.

g> (a.) -

g> (ar) - g) («) ' g> (a,) - g> (^)

4^?3 (^r) - ^2ip Q) - #, + %>' (a*) & (a,)

where

g> (ar) - £> (a.)
Let the constants a be such that

(O - £> (a2)

+

-H...-0

/i equations of which only n — 1 are independent, because the sum of the n
left-hand sides vanishes. Then iu the double summation the coefficient of

i f .1 f u #>' (ar) — &' (z) .
each of the tractions * )—,- — V\ is zero ; and so

and therefore • -^-, = w (w + 1) p (z) + (2n — 1) 2 ^> (a,.).

/IU GLZ" T=l

Hence it follows that

_<T(z + aJ <T(z + a2)...<r(z + an) -z?J("r)
an (z}

satisfies Lame's equation, provided the n constants a be determined by the
preceding equations and by the relation

B = (2n-l) I pfa).

141.] OF LAMP'S EQUATION 285

Evidently the equation is unaltered when — z is substituted for z ; and
therefore

is another solution. Every solution is of the form

MF(z} + NF(-z),
where M and N are arbitrary constants.

COEOLLARY. The simplest cases are when n = l and n = 2.

When n = 1, the equation is

• j-r- + B :

w dzz

there is only a single constant a determined by the single equation

B = p (a),
and the general solution is

,, a (2 + a) ... ,ra(z — a] ... ,
w = M — ^- 7-^-/ e~2£(a) + N - ---- ' ' es^a>
o- (z} a (z)

When n = 2, the equation is

-J-. = 6(0 (z} + B.

w dz*

The general solution is

^

where a and b are determined by the conditions

Rejecting the solution a+b = 0, we have a and b determined by the equations
p (a)

For a full discussion of Lame's equation and for references to the original sources of
information, see Halphen, Traite des fonctions elliptiques, t. ii, chap, xn., in particular,
pp. 495 et seq.

Ex. When Lamp's equation has the form

1 d?w
- -T-5 =n (n + 1) £2 sn20 - h.

w dz2 ^

obtain the solution for w = l, in terms of the Jacobian Theta-Functions,

where co is determined by the equation dn2o> = A-F ; and discuss in particular the solution
when h has the values l+£2, 1, £2.

Obtain the solution for » = 2 in the form

i +B - fe^)e- K& .1

J SI e(») j'

286 PSEUDO-PERIODIC FUNCTIONS [141.

where X and w are given by the equations

(2P sn2 a - 1 - F) (2F sn2 a - 1) (2 sn2 a - 1)
3Fsn4a-2(l+£2)sn2a + l ~ '

and a is derived from h by the relation

Deduce the three solutions that occur when X is zero, and the two solutions that occur
when X is infinite. (Hermite.)

Doubly-Periodic Functions of the Third Kind.

142. The equations characteristic of a doubly-periodic function <I> (z) of
the third kind are

= <£(», <&(z + 2a)') = e~ »~Z Q(z),

where m is an integer different from zero.

Obviously the number of zeros in a parallelogram is a constant, as well as
the number of infinities. Let a parallelogram, chosen so that its sides
contain no zero and no infinity of <& (z}, have p, p + 2<w, p + 2&>' for three
of its angular points; and let a1} a2, . .., a{ be the zeros and cl5 ..., cm be the
infinities, multiplicity of order being represented by repetitions. Then using

"^ (z) to denote , (log <£ (z)}, we have, as the equations characteristic of

*

and for points in the parallelogram

where -ff (^) has no infinity within the parallelogram. Hence

the integral being taken round the parallelogram : by using the Corollary to
Prop. II. in § 116, we have

27ri (I - n) - - \ - \^L\ dz =
Jp \ &> /

so that I = n + m :

or the algebraical excess of the number of irreducible zeros over the number of

irreducible infinities is equal to in.

z
Again, since — = 1 +

z — /A z — p,

a c

we have 2 2 h I — n = z"^ (z) — zH (z),

z — a z — c

and therefore 2-Tn (Sa — 2c) = jz*\? (z) dz,

142.] OF THE THIRD KIND 287

the integral being taken round the parallelogram. As before, this gives

rp+2<a' rp+2<a < vnTri "I

2™ (2a - 2c) = 2ft)^ (z) dz - MV (z) - - (z + 2ft/) dz.

Jp Jp ( ft) )

The former integral is

rp+*»'(g)

,v x dz
(*)

miri

for the side of the parallelogram contains* no zero and no infinity
The latter integral, with its own sign, is

<P(Z) ft)

= 0 + {O + 2« + 2ft>')2 - (p + 2ft/)2}

= 2TO7T* (p + ft) + 2ft)').

Hence 2a — Sc = m (&) + 2&/),

giving the excess of the sum of the zeros over the sum of the infinities in any
parallelogram chosen so as to contain the variable z and to have no one of its
sides passing through a zero or an infinity of the function.

These will be taken as the irreducible zeros and the irreducible infinities :
all others are congruent with them.

All these results are obtained through the theorem II. of § 116, which
assumes that the argument of <y' is greater than the argument of &) or, what
is the equivalent assumption (§ 129), that

rjco' — w'co = ^iri.

143. Taking the function, naturally suggested for the present class by
the corresponding function for the former class, we introduce a function

a(z- d) <r(z- C2). ..<r(z — Cn) '

where the a's and the c's are connected by the relations
Sa — Sc = m (&) + 2&>'), l—n = m.

Then (f>(z) satisfies the equations characteristic of doubly-periodic functions
of the third kind, if

0 = 4Xo) + 2ra77,
k . 27rt = 4X&)2 + 2m?/ft) + 2/ift) + miri — Zmrj (&> + 2ft)') ;

miri — 2mrj' (&> + 2ft)'),

* Both in this integral and in the next, which contain parts of the form I — there is, as in

J w

Prop. VII., § 116, properly an additive term of the form 2iciri, where K is an integer ; but, as there,
both terms can be removed by modification of the position of the parallelogram, and this modifi
cation is supposed, in the proof, to have been made.

288 TERTIARY FUNCTIONS [143.

k and k' being disposable integers. These are uniquely satisfied by taking

with A; = 0, k' = m.

Assuming the last two, the values of X and /JL are thus obtained so as to make
<fr (z) a doubly-periodic function of the third kind.

Now let Oj, ..., di be chosen as the irreducible zeros of <l> (z) and Ci, ..., cn
as the irreducible infinities of <E> (2), which is possible owing to the conditions
to which they were subjected. Then <3> (z)/<j> (z) is a doubly-periodic function
of the first kind; it has no zeros and no infinities in the parallelogram of-
periods and therefore none in the whole plane ; it is therefore a constant, so
that

3> (z) = Ae"** "IZ*+^ - + (l|+8'')} ** <r(*-gi)°-(*-q»)-. •*(*-<*!)

tr(z- d) <r(z- c.2)...o- (z - cn) '

a representation of <3> (z) in terms of known quantities.

Ex. Had the representation been effected by means of the Jacobian Theta-Functions
which would replace a (z) by H(z), then the term in z1 in the exponential would be absent.

144. No limitation on the integral value of m, except that it must not
vanish, has been made : and the form just obtained holds for all values.
Equivalent expressions in the form of sums of functions can be constructed :
but there is then a difference between the cases of m positive and m
negative.

If m be positive, being the excess of the number of irreducible zeros over
the number of irreducible infinities, the function is said to be of positive class
m ; it is evident that there are suitable functions without any irreducible
infinities — they are integral functions.

When m is negative (= — n), the function is said to be of negative class n ;
but there are no corresponding integral functions.

145. First, let m be positive.

i. If the function have no accidental singularities, it can be expressed in
the form

A e**+i* a-(z — a1)a-(z — aa)...<r(z — am),

with appropriate values of X and //..

ii. If the function have n irreducible accidental singularities, then it has
m + n irreducible zeros. We proceed to shew that the function can be
expressed by means of similar functions of positive class m, with a single
accidental singularity.

145.] OF POSITIVE CLASS 289

Using X and /j, to denote

, mri

- 1 — ' and | - - + m (77 + 277'),
&) a)

which are the constants in the exponential factor common to all functions of
the same class, consider a function, of positive class m with a single accidental
singularity, in the form

*m (z, u) = eW ''

<r(u- 6X) o- (u - &„). • • <r (u - bm+1) <r(z-u)'
where b1} b.2, ..., bm are arbitrary constants, of sum s, and
m (&> + 2ft)') = 6OT+1 + fcj + b.> + . . . bm - u

= bm+l +s-u.
The function y-m satisfies the equations

_mirzi

y-w (z + 2<w, u) = i/rm (z, u), y,tt (z + 2&)', w) = e~' « -^m (z, u) ;

regarded as a function of z, it has u for its sole accidental singularity,
evidently simple.

The function - — can be expressed in the form

I\I/* I £ It I

u — k) . . . a- (u — bm) o- {s — m (&)

(r^-b,) ............ a-(z-bm) a{u- z-s + m(a> + 2~w7)} '

Regarded as a function of u, it has z, \, . . ., bm for zeros and z + s - m (to + 2o>')
for its sole accidental singularity, evidently simple : also

z + &J + ...+ bm - {z + s - m (&) + 2o/)} = m (w + 2o>').

Hence owing to the values of X arid p, it follows that -- } - x when re-

f>m(*, tt)

garded as a function of u, satisfies all the conditions that establish a doubly-
periodic function of the third kind of positive class m, so that

1 1

~i 7 ~ =r ^

and therefore

mnz

tym (z, u + 2o>) = ^m (z, u), ^m (z, u + 20)') = e~ijrm (z, u).
Evidently -f m (z, u) regarded as a function of u is of negative class m : its
infinities and its sole zero can at once be seen from the form

-bm) o-{u-z-s+m(ca

<r(u -z)*^-^)...*^- bm) a- {s - m (to + 2o)')j '

Each of the infinities is simple. In the vicinity of u = z, the expansion of
the function is

^^z + positive integral powers of u — z :

19

290 TERTIARY FUNCTIONS [145.

and, in the vicinity of u = br, it is

C* ( 7\

r j + positive integral powers of u — br,

Lv "~~ \Jrp

where Gr (z) denotes

r) <r(z-bi)--.<r(z-br-i)<r(z-br+l)...a(z-bm)o-{z+s-br-m(a>+2a>')}
a- (br - 6j). ..cr (br - 6r_!) cr(br - br+l)...cr(br - bin) o-[s-ra(eo + 2&>')}'

and is therefore an integral function of z of positive class m.

Let 4> (14) be a doubly-periodic function of the third kind, of positive class
m ; and let its irreducible accidental singularities, that is, those which occur
in a parallelogram containing the point u, be a^ of order !+/*!, a., of order
1 + ju,2, and so on. In the immediate vicinity of a point ar, let

--... ±

\

-

rr— r;r-... r-,-~- - -rr.

cm du- du^J u — a,.

Then proceeding as in the case of the secondary doubly-periodic functions
(§ 137), we construct a function

F(u) = 3?(u)^m(z, u).
We at once have F (u + 2o>) = F (u) = F(u + 2a>'),

so that F(u) is a doubly-periodic function of the first kind; hence the sum
of its residues for all the poles in a parallelogram of periods is zero.

For the infinities of F (u), which arise through the factor tym(z, u}, wea
have as the residue for u = z

-<*>(*),
and as the residue for u = br, where r = 1, 2, ..., m,

In the vicinity of a,., we have

fyn (Z, u) = ^rm (Z, «r) + (u - Or) tym' (z, O.r)

where dashes imply differentiation of ^rm {z, u} with regard to u, after which
u is made equal to a,. ; so that in <I> (u) tym (z, u) the residue for u = ar, where
r = l, 2, ..., is

Er (z) = Ar ,jrm (z, ctr) + B, Tjrm' (z, a,.) + Cr tym" (z, ar) + ...+ Mr <^m^r) (z> ar\
Hence we have

and therefore ®(z)= 2 E,(z)+ 2 <& (br) Gr(z),

s=l r=l

giving the expression of <l> (z) by means of doubly -periodic functions of tht
third kind, which are of positive class m and have either no accidental singu->
larity or only one and that a simple singularity.

145.] OF NEGATIVE CLASS 291

The m quantities blt ..., bm are arbitrary; the simplest case which occurs
is when the m zeros of &(z) are different and are chosen as the values
of &!,..., bm. The value of 3>(z) is then

<&(*)= 2 JS'.C*),

s=l

where the summation extends to all the irreducible accidental singularities ;
while, if there be the further simplification that all the accidental singularities
are simple, then

<I> (z) = A1 TJrm (2, «!> + As tym (z, ot2) + . . .,

the summation extending to all the irreducible simple singularities.
The quantity tym (z, ar), which is equal to

) <r(z-bd...<r(z- bm) <r{z + 2b-m(<o + 2ft/) - ar]

a-(ar — b1)...a- (ar - bm) <r {26 - m (co + 2ft>')} a- (z - ar) '

and is subsidiary to the construction of the function E (z\ is called the
simple element of positive class m.

In the general case, the portion

gives an integral function of z, and the portion 2 Es (z) gives a fractional

s=l

function of z.

146. Secondly, let m be negative and equal to — n. The equations
satisfied by & (z} are

i = <I> 0), <I> (z + 2ft)') = e w <£ 0),

and the number of irreducible singularities is greater by n than the number
of irreducible zeros.

One expression for <i> (z} is at once obtained by forming its reciprocal,
which satisfies the equations

11 1 -2-** i

f\ /K / -\ >

and is therefore of the class just considered: the value of is of the

q>(^)

form

ZEs(z) + ^ArGr(z}.

For purposes of expansion, however, this is not a convenient form as it gives
only the reciprocal of <I> (z}.

To represent the function, Appell constructed the element

TT sv°° Ffr-K»-*Wl 7r(2
gr— * . • cot — *-

19—2

292 TERTIARY FUNCTIONS [146.

which, since the real part of to' fan is positive, converges for all values of z and

y, except those for which

z = y (mod. 2&>, 2&>').

For each of these values one term of the series, and therefore the series
itself, becomes infinite of the first order.

Evidently %„ (z, y + 2o>) = %M (z, y},

niryi

Xn (z, y + 2eo') = e ° %„(*, y);
therefore in the present case

0(y)=*(3f)jfr (**?)>

regarded as a function of ^/, is a doubly-periodic function of the first kind.

Hence the sum of the residues of its irreducible accidental singularities
is zero.

When the parallelogram is chosen, which includes z, these singularities
are

(i) y = z, arising through %n (z, y} ;

(ii) the singularities of <£ (y}, which are at least n in number, and are
n + I when <& has I irreducible zeros.

The expansion of Xn 0> y), in powers of y - z, in the vicinity of the point
z, is

+ positive integral powers of y — z ;

y-z
therefore the residue of II (y) is

Let ctr be any irreducible singularity, and in the vicinity of a,, let 3> (y) denote

d

-I- positive integral powers of y — Or,

where the series of negative powers is finite because the singularity is
accidental ; then the residue of H (y} is

Ar ^ (Z, Or) + Br Xn (*, «r) + Cr %,/' (z, Ct,) + . . . + Pr X*™ 0> «>')>

where %n(A) (^, ar) is the value of

dx%n (z, y)

dy*

when y = 0r after differentiation. Similarly for the residues of other singu
larities : and so, as their sum is zero, we have

<£ (Z) = 2 {Ar Xn (*, «r) + Br Xn (*, «•,) + ...+ P, XnW (?, «r)},

the summation extending over all the singularities.

146.] OF NEGATIVE CLASS 293

The simplest case occurs when all the N(>n) singularities a are accidental
and of the first order ; the function 4> (z) can then be expressed in the form

Al Xn (Z, «i) + A2 Xn (Z, Oj) + . . . + AN Xn (z, «#)•

The quantity Xn (z, a), which is equal to

T *^" ^p{(«-i)»'+«} TT 0 - a

a 2/6 COt — -^.

.

2(0

is called the simple element for the expression of a doubly-periodic function of
the third kind of negative class n.

Ex. Deduce the result

_ ^ ( — iVcot
TT snu s=-oov I 2K /'

147. The function Xn (z, y} can be used also as follows. Since Xm (z, y),
qua function of y, satisfies the equations

%m (z, 11 + 2(i)} = Y™ (z, 7/\
llv \ s {/ ' / /V//fc \ J ts /'

miryi

Xm (z, y + 2o/) = e~^xm (z, y),

which are the same equations as are satisfied by a function of y of positive
class m, therefore Xm (<*> z), which is equal to

2 e cot

being a function of z, satisfies the characteristic equations of § 142 ; and, in
the vicinity of z = a,

Xm (a> z) — - — + positive integral powers of z — a.

Z ~~" OC

If then we take the function 4> (z) of § 145, in the case when it has simple
singularities at alt «2, ... and is of positive class m, then

4> (z) + A, xw (a, ,

is a function of positive class m without any singularities: it is therefore
equal to an integral function of positive class m, say to G(z)t where

G (z) = Ae^+^a- (z-al}...(r(z- am),
so that 3>(z) = G(z)-A1Xm(ct1,2)-A,xm(<Xt,z)-....

Ex. As a single example, consider a function of negative class 2, and let it have no
zero within the parallelogram of reference. Then for the function, in the canonical
product-form of § 143, the two irreducible infinities are subject to the relation

and the function is * (z) = AV° "V" -

o- (z— Cj) o- (z-c2)'

294 TERTIARY FUNCTIONS [147.

The simple elements to express 3> (z) as a sum are

2.<!iri , ,

» {{s-lX + Cl} ,77, ' «

*« " «rt (s-C! -2*,),

4iri, ,,
7T -(ci-<o)» - r-w-c'i TT

= _e<-> 2 e a> cot — (2 + 0 j-2no)

after an easy reduction,

4irj

The residue of *(s) for cn which is a simple singularity, is

'Us-(
Al = Ktfa v<

and for c2, also a simple singularity, it is

,

so that ^- = -ew =-ew

^2

Hence the expression for 4> (z) as a sum, which is

!

becomes Al (X2 (2, Cj) - e u ^2 (^ - ci)}

that is, it is a constant multiple of

Again,

— j - -

<r(z- GJ) a- (z + c^ - 2o>-

on changing the constant factor. Hence it is possible to determine L so that

•ni Tti

" C' « c - e<a

Taking the residues of the two sides for z=c1} we have
and therefore finally we have

-C]*- — Ci -- C,

Le™ <» = e °>

>-.•>-*

TtlC

(a (s, c) - e w X2 (2> - c)

<* Cot^L(2-c1-2su)')-e w cot - - (z + cx - 2sw') K

2<a 2a> )

the right-hand side of which admits of further modification if desired.

147.] PSEUDO-PERIODIC FUNCTIONS 295

Many examples of such developments in trigonometrical series are given by Hermite*,
Biehlerf, HalphenJ, Appell§, and Krause||.

148. We shall not further develop the theory of these uniform doubly-
periodic functions of the third kind. It will be found in the memoirs of
Appell§ to whom it is largely due; and in the treatises of Halphen**, and
of Rausenberger"f"f.

It need hardly be remarked that the classes of uniform functions of a
single variable which have been discussed form only a small proportion of
functions reproducing themselves save as to a factor when the variable
is subjected to homographic substitutions, of which a very special example
is furnished by linear additive periodicity. Thus there are the various
classes of pseudo-automorphic functions, (§ 305) called Thetafuchsian by Pom-
care, their characteristic equation being

for all the substitutions of the group determining the function : and other
classes are investigated in the treatises which have just been quoted.

The following examples relate to particular classes of pseudo-periodic
functions.

Ex. 1. Shew that, if F (z) be a uniform function satisfying the equations

m

where b is a primitive mth root of unity, then F(z) can be expressed in the form

where f(z) denotes the function

and prove that \F(z)dz can be expressed in the form of a doubly-periodic function
together with a sum of logarithms of doubly-periodic functions with constant coefficients.

(Goursat.)

* Comptes Rendus, t. Iv, (1862), pp. 11—18.

t Sur les developpements en series des fonctions doublement periodiqucs de troisieme espece,
(These, Paris, Gauthier-Villars, 1879).

£ Traite des fonctions elliptiques, t. i, chap. xm.

§ Annales de VEc. Norm. Sup., 3rae S6r., t. i, pp. 135—164, t. ii, pp. 9—36, t. iii, pp. 9—42.

|| Math. Ann., t. xxx, (1887), pp. 425—436, 516—534.

'* Traite des fonctions elliptiques, t. i, chap. xiv.

ft Lehrbuch der Theorie der periodischen Functional, (Leipzig, Teubner, 1884), where further
references are given.

296 PSEUDO-PERIODIC FUNCTIONS [148.

Ex. 2. Shew that, if a pseudo-periodic function be denned by the equations

and if, in the parallelogram of periods containing the point z, it have infinities c, ... such
that in their immediate vicinity

then/ (2) can be expressed in the form

-'^'^«{^I+ ...... +«»,£}«—>,

the summation extending over all the infinities of/ (z) in the above parallelogram of periods,
and the constants (715 ... being subject to the condition

+ iVS Cl = A o>' — X'«o.

Deduce an expression for a doubly-periodic function <f) (z) of the third kind, by
assuming

/W-f]8. (Halphen.)

(f> \g)

Ex. 3. If S(z) be a given doubly-periodic function of the first kind, then a
pseudo-periodic function F(z), which satisfies the equations

F(z + ^} = F(z),
mriz
F (z + 2o>') = e ~"~ S (z} F (z),

where n is an integer, can be expressed in the form

where -4 is a constant and TT (2) denotes

the summation extending over all points &,. and the constants Br being subject to the
relation

Explain how the constants b, G and B can be determined. (Picard.)

Ex. 4. Shew that the function F(z) defined by the equation

for values of \z\, which are <1, satisfies the equation

and that the function Fl(a!)=^ ^rjr-£i

where (j)(,v) = 3? — 1, and </>„(.*•')> f()r positive and negative values of n, denotes (/> [0 {<£ 0 (#)}]

<f> being repeated n times, and a is the positive root of a3 — a - 1 = 0 ; satisfies the equation

for real values of the variable.

Discuss the convergence of the series which defines the function Fl (x). (Appell.)

CHAPTER XIII.

FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM.

149. WE may consider at this stage an interesting set* of important
theorems, due to Weierstrass, which are a justification, if any be necessary,
for the attention ordinarily (and naturally) paid to functions belonging to
the three simplest classes of algebraic, simply-periodic and doubly-periodic
functions.

A function <f> (u) is said to possess an algebraical addition theorem, when
among the three values of the function for arguments u, v, and u + v, where u
and v are general and not merely special arguments, an algebraical equation
exists f having its coefficients independent of u and v.

150. It is easy to see, from one or two examples, that the function does
not need to be a uniform function of the argument. The possibility of
multiformity is established in the following proposition :

A function defined by an algebraical equation, the coefficients of which are
uniform algebraical functions of the argument, or are uniform simply -periodic
functions of the argument, or are uniform doubly -periodic functions of the
argument, possesses an algebraical addition-theorem.

* They are placed in the forefront of Schwarz's account of Weierstrass's theory of elliptic
functions, as contained in the Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen;
but they are there stated (§§ 1—3) without proof. The only proof that has appeared is in a
memoir by Phragmen, Acta Math., t. vii, (1885), pp. 33—42; and there are some statements
(pp. 390—393) in Biermann's Theorie der analytischen Functionen relative to the theorems. The
proof adopted in the text does not coincide with that given by Phragme'n.

t There are functions which possess a kind of algebraical addition -theorem ; thus, for
instance, the Jacobian Theta-functions are such that eA(u + w) O^ (u- v) can be rationally ex
pressed in terms of the Theta-functions having it and v for their arguments. Such functions
are, however, naturally excluded from the class of functions indicated in the definition.

Such functions, however, possess what may be called a multiplication-theorem for multipli
cation of the argument by an integer, that is, the set of functions 6 (nut) can be expressed
algebraically in terms of the set of functions 6 (M). This is an extremely special case of a set
of transcendental functions having a multiplication-theorem, which are investigated by Poincare,
Liouville, 4°" S6r., t. iv, (1890), pp. 313—365.

298 EXAMPLES OF FUNCTIONS [150.

First, let the coefficients be algebraical functions of the argument u. If
the function defined by the equation be U, we have

Umg0 (u) + Um~lgi (u) + ...+gm (u) = 0,

where g0(u),gi(u}, ...,gm(u) are rational integral algebraical functions of u
of degree, say, not higher than n. The equation can be transformed into

un f/U\+ u'1-1/! ( U) + ... + fn ( U) = 0,

where f0(U), fi(U), ••••> fn(U) are rational integral algebraical functions of
U of degree not higher than m.

If V denote the function when the argument is v, and W denote it when
the argument is u + v, then

w»/0 (7) + ^1/1 (7) + ... +fn (V) M 0,
and (u + v)n/0 ( W) + (u + vY^f, ( W ) + . . . +fn ( W ) = 0.

The algebraical elimination of the two quantities u and v between these
three equations leads to an algebraical equation between the quantities
/(£/"), /(7) and f (W), that is, to an algebraical equation between U, V, W,
say of the form

G(U, V, F) = 0,

where G denotes an algebraical function, with coefficients independent of
u and v. It is easy to prove that G is symmetrical in U and 7, and that
its degree in each of the three quantities U, 7, W is wn2. The equation
G = 0 implies that the function U possesses an algebraical addition- theorem.

Secondly, let the coefficients* be uniform simply-periodic functions of
the argument u. Let &> denote the period: then, by § 113, each of these

TT'IL

functions is a rational algebraical function of tan — . Let u' denote

tan — ; then the equation is of the form

Umg0 (u') + Um^g, (u'} + ...+ gm 00 = 0,

where the coefficients g are rational algebraical (and can be taken as
integral) functions of u'. If p be the highest degree of u' in any of them,
then the equation can be transformed into

u'vfo ( U) + u'P-1/! ( U) + . . . + fp ( U) = 0,

where f0(U), fi(U), ..., fp(U) are rational integral algebraical functions of
U of degree not higher than m.

* The limitation to uniformity for the coefficients has been introduced merely to make the
illustration simpler; if in any case they were multiform, the equation would be replaced by
another which is equivalent to all possible forms of the first arising through the (finite)
multiformity of the coefficients : and the new equation would conform to the specified
conditions.

150.] POSSESSING AN ADDITION-THEOREM 299

Let v denote tan — , and w denote tan — -- ; then the corresponding
cy &)

values of the function are determined by the equations

and w'*>f0(W) + w'p-*/! (W) + ... +fp (W) = 0.

The relation between u', v', w' is

u'v'w' + u' + v' - w' = 0.

The elimination of the three quantities u', v', w' among the four equations
leads as before to an algebraical equation

G(U, V, W) = 0,

where G denotes an algebraical function (now of degree mp'2) with coefficients
independent of u and v. The function U therefore possesses an algebraical
addition-theorem.

Thirdly, let the coefficients be uniform doubly-periodic functions of the
argument u. Let &> and &/ be the two periods ; and let @ (u), the Weier-
strassian elliptic function in those periods, be denoted by £. Then every
coefficient can be expressed in the form

~L '

where L, M, N are rational integral algebraical functions of f of finite
degree. Unless each of the quantities N is zero, the form of the equation
when these values are substituted for the coefficients is

A+Bp'(u) = 0,

so that A* = &(±?-g£-9*)\

and this is of the form

Umff* (£) + U'^g, (|) + . . . + gm (£) - 0,

where the coefficients g are rational algebraical (and can be taken as integral)
functions of £ If q be the highest degree of £ in any of them, the equation
can be transformed into

where the coefficients / are rational integral algebraical functions of U of
degree not higher than 2m.

Let TJ denote $ (v) and f denote p(u + v); then the corresponding values
of the function are determined by the equations

......... +fq(V)=0,

By using Ex. 4, § 131, it is easy to shew that the relation between £, rj, £ is

300 WEIERSTRASS'S THEOREM ON FUNCTIONS [150.

The elimination of £, ij, £ from the three equations leads as before to an

algebraical equation

G(U,V, W) = 0,

of finite degree and with coefficients independent of u and v. Therefore in this
case also the function U possesses an algebraical addition-theorem.

If, however, all the quantities N be zero, the equation defining U is of the

form

Umh0 (£) + U^h, (£) + . . . + hm (£) = 0,

and a similar argument then leads to the inference that U possesses an
algebraical addition-theorem.

The proposition is thus completely established.

151. The generalised converse of the preceding proposition now suggests
itself : what are the classes of functions of one variable that possess an alge
braical addition-theorem? The solution is contained in Weierstrass's theorem : —

An analytical function <f> (u), which possesses an algebraical theorem, is
either

(i) an algebraical function of u ; or

liru

(ii) an algebraical function of e » , where w is a suitably chosen
constant ; or

(iii) an algebraical function of the elliptic function %>(u), the periods — or
the invariants g.z and g3 — being suitably chosen constants.

Let U denote </> (w).

For a given general value of u, the function U may have m values where,
for functions in general, there is not a necessary limit to the value of m ; it
will be proved that, when the function possesses an algebraical addition-
theorem, the integer m must be finite.

For a given general value of U, that is, a value of U when its argument is
not in the immediate vicinity of a branch-point if there be branch-points, the
variable u may have p values, where p may be finite or may be infinite.

Similarly for given general values of v and of V, which will be used to
denote <£ (v).

First, let p be finite. Then because u has p values for a given value of U
and v has p values for a given value of V, and since neither set is affected by the
value of the other function, the sum u + v has p2 values because any member of
the set u can be combined with any member of the set v ; and this number
p2 of values of u + v is derived for a given value of U and a given value of V.

Now in forming the function <j>(u + v), which will be denoted by W, we
have m values of W for each value of u + v and therefore we have mp2 values
of W for the whole set, that is, for a given value of U and a given value of V.

151.] POSSESSING AN ADDITION-THEOREM 301

Hence the equation between U, V, W is of degree* mp2 in W, necessarily
finite when the equation is algebraical ; and therefore m is finite.

Because m is finite, U has a finite number m of values for a given value of
u ; and, because p is finite, u has a finite number p of values for a given value of
U. Hence U is determined in terms of u by an algebraical equation of degree
m, the coefficients of which, are rational integral algebraical functions of
degree p ; and therefore U is an algebraic function of u.

152. Next, let p be infinite ; then (see Note, p. 303) the system of values
may be composed of (i) a single simply-infinite series of values or (ii) a finite
number of simply-infinite series of values or (iii) a simply-infinite number of
simply-infinite series of values, say, a single doubly-infinite series of values or
(iv) a finite number of doubly-infinite series of values or (v) an infinite
number of doubly-infinite series of values where, in (v), the infinite number
is not restricted to be simply-infinite.

Taking these alternatives in order, we first consider the case where the p
values of u for a given general value of U constitute a single simply -infinite
series. They may be denoted by f (u, n), where n has a simply-infinite
series of values and the form of/ is such that f(u, 0) = u.

Similarly, the p values of v for a given general value of V may be denoted
by/(y, n), where n' has a simply-infinite series of values. Then the different
values of the argument for the function W are the set of values given by

f(u,n)+f(v,ri),

for the simply-infinite series of values for n and the similar series of values
for n'.

The values thus obtained as arguments of W must all be contained in
the series f(u + v, n"}, where n" has a simply-infinite series of values ; and,
in the present case,/(w + w, n"} cannot contain other values. Hence for some
values of n and some values of n', the total aggregate being not finite, the
equation

f(u,n}+f(v,n'}=f(u + v,n")
must hold, for continuously varying values of u and v.

In the first place, an interchange of u and v is equivalent to an interchange
of n and n on the left-hand side; hence n" is symmetrical in n and n'.
Again, we have

df(u, n) _ df(u + v, n")
du 3 (u + v)

dv '

* The degree for special functions may be reduced, as in Cor. 1, Prop. XIII, § 118; but in no
case is it increased. Similarly modifications, in the way of finite reductions, may occur in the
succeeding cases ; but they will not be noticed, as they do not give rise to essential modification
in the reasoning.

302 FORM OF ARGUMENT [152.

so that the form of f(u, n) is such that its first derivative with regard to u is
independent of u. Let 0 (n) be this value, where 0 (n), independent of u, may
be dependent on n ; then, since

we have f(u, n) = uO (n) + ty (n),

-fy- (n) being independent of u. Substituting this expression in the former

equation, we have the equation

u6 (n) + ^ (n) + v9 (n'} + f (71') = (u + v)6 (n"} + ^ (n"),
which must be true for all values of u and v ; hence

e(n)=e(n") = d(n'),

so that 6 (n) is a constant and equal to its value when n = 0. But when n is
zero,/(w, 0) is u ; so that 9 (0) = 1 and ^ (0) = 0, and therefore

f(u, n) = u + Tjr (n),
where i/r vanishes with n.

The equation defining ty is

for values of n from a singly-infinite series and for values of n' from the same
series, that series is reproduced for TO". Since ^ (n) vanishes with n, we take

^ (n) = HX (n),

and therefore rc% (n) + n'% (n') = ri'x (n").

Again, when n' vanishes, the required series of values of n" is given by taking
n" = n ; and, when n does not vanish, n" is symmetrical in n and n', so that

we have

n" = n + n' + nn\,

where X is not infinite for zero or finite values of n or n'. Thus
•HX (n) + n'x (n) = (n + TO' + -nw'X) % (w + ?*' + wi'X).

Since the left-hand side is the sum of two functions of distinct and inde
pendent magnitudes, the form of the equation shews that it can be satisfied

only if

X = 0, so that n" = n + n' ;

and % 0) = % (n//)

= %(n'\
so that each is a constant, say o> ; then

f(u, n} = u + nco,

which is the form that the series must adopt when the series f(u + v, n") is
obtained by the addition of/(«, n) and/0, n')-

152.] IN A SIMPLY-INFINITE SERIES 303

It follows at once that the single series of arguments for W is obtained,
as one simply-infinite series, of the form u + v+n"a). For each of these
arguments we have m values of W, and the set of m values of W is
the same for all the different arguments; that is, W has m values for a
given value of U and a given value of V. Moreover, U has m values for each
argument and likewise V; hence, as the equation between U, V, W is of
a degree that is necessarily finite because the equation is algebraical, the
integer m is finite.

It thus appears that the function U has a finite number m of values for
each value of the argument u, and that for a given value of the function the
values of the argument form a simply-periodic series represented by u + nw.

But the function tan ( — ) is such that, for a given value, the values of the

V 03 J

argument are represented by the series u + nw ; hence for each value of

tan ( — 1 there are m values of U and for each value of U there is one value
\ «o /

of tan -- . It therefore follows, by SS 113, 114, that between U and tan (— }
w \ to /

there is an algebraical relation which is of the first degree in tan - - and the

O)

U

rath degree in U, that is, U is an algebraic function of tan — - . Hence U is

(I)

an algebraic function also of e <" .

Note. This result is based upon the supposition that the series of argu
ments, for which a branch of the function has the same value, can be arranged
in the form/(w, n), where n has a simply-infinite series of integral values. If,
however, there were no possible law of this kind — the foregoing proof shews
that, if there be one such law, there is only one such law, with a properly
determined constant co — then the values would be represented by ul} u», ...,up
with p infinite in the limit. In that case, there would be an infinite number of
sets of values for u + v of the type WA + v^, where X and p might be the same
or might be different ; each set would give a branch of the function W and then
there would be an infinite number of values of W corresponding to one branch
of U and one branch of V. The equation between U, V and W would be of
infinite degree in W, that is, it would be transcendental and not algebraical.
The case is excluded by the hypothesis that the addition-theorem is alge
braical, and therefore the equation between U, V and W is algebraical.

153. Next, let there be a number of simply-infinite series of values of
the argument of the function, say q, where q is greater than unity and
may be either finite or infinite. Let ul} u.2, ..., uq denote typical members
of each series.

Then all the members of the series containing ul must be of the form

304 FORM OF ARGUMENT [153.

fi (ui> n)> f°r an infinite series of values of the integer n. Otherwise, as in the
preceding note, the sum of the values in the series of arguments u and of
those in the same series of arguments v would lead to an infinite number of
distinct series of values of the argument u + v, with a corresponding infinite
number of values W ; and the relation between U, V, W would cease to be
algebraical.

In the same way, the members of the corresponding series containing ^
must be of the form/! (v1} ri) for an infinite series of values of the integer n'.
Among the combinations

the simply-infinite series fi(tii+v1} n") must occur for an infinite series
of values of n"; and therefore, as in the preceding case,

fi(uly n) = M1 + nw1,

where toj is an appropriate constant. Further, there is only one series of
values for the combination of these two series ; it is represented by

Ui + v1 + n"wl.

In the same way, the members of the series containing u2 can be repre
sented in the form u2 + nco2, where o>2 is an appropriate constant, which may
be (but is not necessarily) the same as Wj ; and the series containing u.2,
when combined with the set containing v2, leads to only a single series
represented in the form u.2 + v2 + ri'o)2. And so on, for all the series in order.

But now since u2 + m2a)2, where m2 is an integer, is a value of u for a given
value of U, it follows that U (u2 + ra2a>2) = U (w2) identically, each being equal

to U. Hence

U (M! + mlwl + 7n.,<y2) = U (i^ + ra^) = U (u^ = U,

and therefore ^ + ml(al + ra2&>2 is also a value of u for the given value of U,
leading to a series of arguments which must be included among the original
series or be distributed through them. Similarly u1 + 2mr(i)r, where the
coefficients ra are integers and the constants to are properly determined,
represents a series of values of the variable u, included among the original
series or distributed through them. And generally, when account is taken of
all the distinct series thus obtained, the aggregate of values of the variable u
can be represented in the form Wx+2wrtur, for \ — 1, 2, ..., K, where K is
some finite or infinite integer.

Three cases arise, (a) when the quantities « are equal to one another or
can be expressed as integral multiples of only one quantity a>, (6) when the
quantities &> are equivalent to two quantities f^ and O2 (the ratio of which is
not real), so that each quantity &> can be expressed in the form

a>r=plrfil+parsia>

the coefficients plr, p2r being finite integers ; (c) when the quantities « are
not equivalent to only two quantities, such as flj and fl2.

153.] SIMPLY-PERIODIC FUNCTIONS 305

For case (a), each of the K infinite series of values u can be expressed
in the form u^+pci), for X = 1, 2, ..., « and integral values of p.

First, let K be finite, so that the original integer q is finite. Then the
values of the argument for W are of the type

that is, MA + '?V +£>"&>,

for all combinations of \ and fju and for integral values of p". There are thus
K- series of values, each series containing a simply-infinite number of terms
of this type.

For each of the arguments in any one of these infinite series, W has ra
values ; and the set of m values is the same for all the arguments in one and
the same infinite series. Hence W has w/c2 values for all the arguments in
all the series taken together, that is, for a given value of U and a given
value of V. The relation between U, V, W is therefore of degree m«2,
necessarily finite when the equation is algebraical ; hence m is finite.

It thus appears that the function U has a finite number m of values for
each value of the argument u, and that for a given value of the function there
are a finite number K of distinct series of values of the argument of the form

7TU

u+poi), w being the same for all the series. But the function tan -- has
one value for each value of u and the series u+pat represents the series of

7TU

values of u for a given value of tan — . It therefore follows that there are

CO

m values of U for each value of tan — and that there are K values of tan —

to o>

for each value of U ; and therefore there is an algebraical relation between

U and tan — , which is of degree K in the latter and of degree m in the
&)

iiru
TTlI

former. Hence U is an algebraic function of tan — and therefore also of e M .

Next, let K be infinite, so that the original integer q is infinite. Then,
as in the Note in § 152, the equation between U, V, W will cease to be
algebraical unless each aggregate of values u^+pw, for each particular
value of p and for the infinite sequence X= 1, 2, ..., K, can be arranged in a
system or a set of systems, say a in number, each of the form fp(u+pa), pp)
for an infinite series of values of pp. Each of these implies a series of values
fp(v+p'u>, pp) of the argument of V for the same series of values of pp as of
pp> and also a series of values fp(u + v+p"(o, pp") of the argument of W for
the same series of values of pp". By proceeding as in § 152, it follows that

fp (u +pa>, pp} = u+pto +pp(0p,

where &>p' is an appropriate constant, the ratio of which to &> can be proved
F. 20

306 FORM OF ARGUMENT [153.

(as in § 106) to be not purely real, and pp has a simply-infinite succession of
values. The integer a may be finite or it may be infinite.

When ay and all the constants o>' which thus arise are linearly equivalent
to two quantities f^ and O2, so that the terms additive to u can be expressed
in the form 8^ + s.2fl», then the aggregate of values u can be expressed
in the form

for a simply-infinite series for pl and for p2 ; and p has a series of values
1, 2, ..., <r. This case is, in effect, the same as case (6).

When o) and all the constants «' are not linearly equivalent to only
two quantities, such as Oj and IL>, we have a case which, in effect, is the
same as case (c).

These two cases must therefore now be considered.

For case (6), either as originally obtained or as derived through parfc
of case (a), each of the (doubly) infinite series of values of u can be expressed
in the form

for X = 1, 2, ..., <r and for integral values of _p, and p,. The integer a may be
finite or infinite ; the original integer q is infinite.

First, let cr be finite. Then the values of the argument for W are of the
type

that is, u\ + v^ +pi"£li + p2"O2,

for all combinations of \ and p and for integral values of £>/' and p.". There
are thus cr2 series of values, each series containing a doubly-infinite number ofl
terms of this type.

For every argument there are m values of W ; and the set of m values is
the same for all the arguments in one and the same infinite series. Thus W
has mo-2 values for all the arguments in all the series, that is, for a given value
of U and a given value of V; and it follows, as before, from the consideration i
of the algebraical relation, that m is finite.

The function U thus has m values for each value of the argument u ; and
for a given value of the function there are cr series of values of the argument,
each series being of the form wx + PI^I +p.2Q*-

Take a doubly-periodic function © having Oj and H2 for its periods, such*1
that for a given value of © the values of its arguments are of the foregoing
form. Whatever be the expression of the function, it is of the order cr. ,
Then U has m values for each value of @, and @ has one value for each'.
value of U; hence there is an algebraical equation between U and ©, ow

* All that is necessary for this purpose is to construct, by the use of Prop. XII, § 118, ai
function having, as its irreducible simple infinities, a series of points aj, a2,..., a<7 — special*
values of «j, w2, ..., ua— in the parallelogram of periods, chosen so that no two of the <r points a
coincide.

153.] DOUBLY-PERIODIC FUNCTIONS 307

:he first degree in the latter and of the rath degree in U: that is, U is an
algebraical function of @. But, by Prop. XV. § 119, © can be expressed in
the form

where L, M, N are rational integral algebraical functions of $ (u), if f^ and H2
be the periods of g) (u); and g)' (u) is a two- valued algebraical function of jjp (u),
so that © is an algebraical function of i@ (u). Hence also U is an algebraical
function of $(u\ the periods o/<p (u) being properly chosen.

This inference requires that a, the order of ©, be greater than 1.
Because U has m values for an argument u, the symmetric function St/"
has one value for an argument u and it is therefore a uniform function.
But each term of the sum has the same value for u+pifli+pflt as for
u ; and therefore this uniform function is doubly-periodic. The number of
independent doubly-infinite series of values of u for a uniform doubly-
periodic function is at least two : and therefore there must be at least two
doubly-infinite series of values of u, so that <r > 1. Hence a function, that
possesses an addition-theorem, cannot have only one doubly-infinite series of
values for its argument.

If cr be infinite, there is an infinite series of values of u of the form
+ p^ + p.flz ; an argument, similar to that in case (a), shews that this is,
in effect, the same as case (c).

It is obvious that cases (ii), (iii) and (iv) of § 152 are now completely
covered ; case (v) of § 152 is covered by case (c) now to be discussed in § 154.

154. For case (c), we have the series of values u represented by a number
of series of the form

where the quantities &> are not linearly equivalent to two quantities flj and
Q2- The original integer q is infinite.

Then, by §§ 108, 110, it follows that integers m can be chosen in an
unlimited variety of ways so that the modulus of

r=l

is infinitesimal, and therefore in the immediate vicinity of any point u^
there is an infinitude of points at which the function resumes its value.
Such a function would, as in previous instances, degenerate into a mere
constant ; and therefore the combination of values which gives rise to this
case does not occur.

All the possible cases have been considered: and the truth of Weierstrass's

20—2

308 EXAMPLES [154.

theorem* that a function, which has an algebraical addition-theorem, is either

imi

an algebraical function of u, or of e " (where &> is suitably chosen), or of g> (u),
where the periods of @(u) are suitably chosen, is established; and it has
incidentally been established — it is, indeed, essential to the derivation of the
theorem — that a function, which has an algebraical addition-theorem, has only
a finite number of values for a given argument.

It is easy to see that the first derivative has only a finite number of values
for a given argument; for the elimination of U between the algebraical
equations

, ,

leads to an equation in U' of the same finite degree as G in U.

Further, it is now easy to see that if the analytical function <£ (u), which
possesses an algebraical addition-theorem, be uniform, then it is a rational

iiru

function either of u, or of e w , or of $> (u) and $' (u) ; and that any uniform
function, which is transcendental in the sense of § 47 and which possesses an
algebraical addition-theorem, is either a simply-periodic function or a doubly-
periodic function.

The following examples will illustrate some of the inferences in regard to the number
of values of <p (u + v) arising from series of values for u and v.

Ex. I. Let U=u* + (2u+l)*.

Evidently m, the number of values of U for a value of u, is 4 ; and, as the rationalised
form of the equation is

the value of p, being the number of values of u for a given value of U, is 2. Thus the
equation in W should be, by § 151, of degree (4.22 — ) 16.

This equation is n {3 ( W2 - U2 - F2) + 1 - 2kr} = 0,

HI
where kr is any one of the eight values of

W(2W*-I)*+U(2U*-l$+V(2V*-l)*;

' •

an equation, when rationalised, of the 16th degree in W.

Ex. 2. Let U=cosu.

Evidently m = l; the values of u for a given value of U are contained in the double
series u + 2irn, -u + 2irn, for all values of n from -QO to +GO. The values of u + v are
, that is, u + v + 27rp; -u + 27rn+v + 2irm, that is, -u + v + 2-n-p ;
, that is, u-v + ^Trp; -u + 2irn-v + 2irm, that is, -u-v + Znp,

* The theorem has been used by Schwarz, Ges. Werke, t. ii, pp. 260—268, in determining all
the families of plane isothermic cirrves which are algebraical curves, an 'isothermic' curve being
of the form u = c, where w is a function satisfying the potential-equation

154.] THE DIFFERENTIAL EQUATION 309

to that the number of series of values of u+v is four, each series being simply-infinite.
It might thus be expected that the equation between U, V, W would be of degree

4 = ) 4 in W ; but it happens that

cos (u + v)=cos( -u-v),
and so the degree of the equation in W is reduced to half its degree. The equation is

W2 - 2 WU V+ U2 + V2 - 1 = 0.

Ex. 3. Let U=&iiu.

Evidently m = l; and there are two doubly-infinite series of values of u determined
by a given value of U, having the form u + 2ma> + 2m'<o', o> - w + 2mo> + 2m V. Hence the
values of u + v are

= u+v (mod. 2c0, 2o>') ; = ca-u + v (mod. 2«, 2«') ;
= ca + u-v(mod. 2o>, 2<o') ; = -u-v (mod. 2o>, 2&>') ;

four in number. The equation may therefore be expected to be of the fourth degree
in W; it is

4 (1 - 6T2) (1 - F2) (1 - IF2) = (2 - U2- F2- IF2 +£2*7272 W2^

155. But it must not be supposed that any algebraical equation between
U, V, W, which is symmetrical in U and V, is one necessarily implying the
representation of an algebraical addition-theorem. Without entering into a
detailed investigation of the formal characteristics of the equations that are
suitable, a latent test is given by implication in the following theorem, also
due to Weierstrass : —

If an analytical function possess an algebraical addition-theorem, an
algebraical equation involving the function and its first derivative with regard
to its argument exists ; and the coefficients in this equation do not involve the
argument of the function.

The proposition might easily be derived by assuming the preceding
proposition, and applying the known results relating to the algebraical
dependence between those functions, the types of which are suited to the
representation of the functions in question, and their derivatives ; we shall,
however, proceed more directly from the equation expressing the algebraical
addition-theorem in the form

G(U,V, F) = 0,

which may be regarded as a rationally irreducible equation.
Differentiating with regard to u, we have

WU'+MW^Q

dUL +dW '
and similarly, with regard to v, we have

a>+ *<=<>,

from which it follows that

310 EXPRESSION OF [155.

This equation* will, in general, involve W; in order to obtain an equation
free from W, we eliminate W between

n A a ^^ rr/ d6r Tr/

G = 0 and ^j- U' = „ V ,

oil ov

the elimination being possible because both equations are of finite degree;
and thus in any case we have an algebraical equation independent of W and
involving U, U', V, V.

Not more than one equation can arise by assigning various values to v, a
quantity that is independent of u ; for we should have either inconsistent
equations or simultaneous equations which, being consistent, determine a!
limited number of values of U and U' for all values of u, that is, only a
number of constants. Hence there can be only one equation, obtained by
assigning varying values to v; and this single equation is the algebraical
equation between the function and its first derivative, the coefficients being
independent of the argument of the function.

Note. A test of suitability of an algebraical equation G — 0 between
three variables U, V, W to represent an addition-theorem is given by the
condition that the elimination of W between

G-Q and U'^-V —

dU~ dV

leads to only a single equation between U and U' for different values of V
and V.

Ex. Consider the equation

(Z-U- V- W)*-4(1-U}(1- F)(l- F) = 0.
The deduced equation involving U1 and V is

(2FTF- V- W+ U} U' = (2UW- U- W+ V) V,

, th-it W (V-U}(V'+U'}

= (SV~lTUr

The elimination of W is simple. We have

_

(27-1) U'-(2U-\) F"

F U'-l-U V'

utd 2 U V W-«

(

Neglecting 4 (F+ U— 1) = 0, which is an irrelevant equation, arid multiplying by
(2F— 1) U' — (2U—l) F', which is not zero unless the numerator also vanish, and this
would make both U' and V zero, we have

( F+ U- 1) {(1 - F) U' - (1 - U} F'} 2 = (1 - U) (1 - F) ( U' - F') (2 F- 1) U' - (2 U- 1) F'},
and therefore V(U-V}(1- V] (7'2+ U( F- U} (1 - U} F'2 = 0.

It is permissible to adopt any subsidiary irrational or non-algebraical form as the equivalent
of G = 0, provided no special limitation to the subsidiary form be implicitly adopted. Thus, if W
can be expressed explicitly in terms of U and F, this resoluble (but irrational) equivalent of the
equation often leads rapidly to the equation between U and its derivative.

155.] THE ADDITION-THEOREM 311

When the irrelevant factor U- V is neglected, this equation gives

U'* F'2

U(l-U}~ V(l - V) '

the equation required : and this, indeed, is the necessary form in which the equation
involving U and U' arises in general, the variables being combined in associate pairs.
Each side is evidently a constant, say 4a2 ; and then we have

Then the value of U is sin2 (aM+/3), the arbitrary additive constant of integration
being /3 ; by substitution in the original equation, (3 is easily proved to be zero.

156. Again, if the elimination between

a - o and — U' - — V

aduu ~wv

be supposed to be performed by the ordinary algebraical process for finding

o/~y o/^r

the greatest common measure of G and U' %Tf — V %-\r> regarded as functions

of W, the final remainder is the eliminant which, equated to zero, is the
differential equation involving U, U', V, F'; and the greatest common measure,
equated to zero, gives the simplest equation in virtue of which the equations

G = 0 and ^y U' = _-^ V subsist. It will be of the form
oil ov

f(W,U,V, U',V') = 0.

If the function have only one value for each value of the argument, so that it
is a uniform function, this last equation can give only one value for W', for all
the other magnitudes that occur in the equation are uniform functions of
their respective arguments. Since it is linear in W, the equation can be
expressed in the form

W = R(U, V, U', V'\

where R denotes a rational function. Hence* : —

A uniform analytical function (f> (u), which possesses an algebraical
addition-theorem, is such that (f> (u + v) can be expressed rationally in terms
of $ (u), <£' (w), $ (v) and <j> (v).

It need hardly be pointed out that this result is not inconsistent with the
fact that the algebraical equation between (£ (u + v), (f> (u) and <f> (v) does not,
in general, express $(u + v) as a rational function of (f> (u) and <f>(v). And it
should be noticed that the rationality of the expression of <£ (u + v) in terms
of <j) (u), $ (v), (/>' (w), $ (v) is characteristic of functions with an algebraical
addition-theorem. Instances do occur of functions such that <j)(u + v) can be
expressed, not rationally, in terms of <£ (u), </> (v), </>' (u), </>' (v) ; they do not
possess an algebraical addition-theorem. Such an instance is furnished by
%(u)', the expression of £(u + v), given in Ex. 3 of § 131, can be modified so '
as to have the form indicated.

* The theorem is due to Weierstrass ; see Schwarz, § 2, (I.e. in note to p. 297).

CHAPTER XIV.

CONNECTION OF SURFACES.

157. IN proceeding to the discussion of multiform functions, it was
stated (§ 100) that there are two methods of special importance, one of which
is the development of Cauchy's general theory of functions of complex vari
ables and the other of which is due to Riemann. The former has been
explained in the immediately preceding chapters ; we now pass to the
consideration of Riemann's method. But, before actually entering upon it,
there are some preliminary propositions on the connection of surfaces which
must be established ; as they do not find a place in treatises on geometry, an
outline will be given here but only to that elementary extent which is
necessary for our present purpose.

In the integration of meromorphic functions, it proved to be convenient
to exclude the poles from the range of variation of the variable by means of
infinitesimal closed simple curves, each of which was thereby constituted a
limit of the region : the full boundary of the region was composed of the
aggregate of these non-intersecting curves.

Similarly, in dealing with some special cases of multiform functions, it
proved convenient to exclude the branch-points by means of infinitesimal
curves or by loops. And, in the case of the fundamental lemma of § 16, the
region over which integration extended was considered as one which possibly
had several distinct curves as its complete boundary.

These are special examples of a general class of regions, at all points
within the area of which the functions considered are monogeiiic, finite, and
continuous and, as the case may be, uniform or multiform. But, important
as are the classes of functions which have been considered, it is necessary to
consider wider classes of multiform functions and to obtain the regions which
are appropriate for the representation of the variation of the variable in each
case. The most conspicuous examples of such new functions are the algebraic
functions, adverted to in §§ 94 — 99 ; and it is chiefly in view of their value
and of the value of functions dependent upon them, as well as of the kind of
surface on which their variable can be simply represented, that we now
proceed to establish some of the topological properties of surfaces in general.

158. A surface is said to be connected when, from any point of it to any
other point of it, a continuous line can be drawn without passing out of the

158.]

EXAMPLES OF CONNECTED SURFACES

313

surface. Thus the surface of a circle, that of a plane ring such as arises in
Lambert's Theorem, that of a sphere, that of an anchor-ring, are connected
surfaces. Two non-intersecting spheres, not joined or bound together in any
manner, are not a connected surface but are two different connected surfaces.
It is often necessary to consider surfaces, which are constituted by an
aggregate of several sheets ; but, in order that the surface may be regarded
as connected, there must be junctions between the sheets.

One of the simplest connected surfaces is such a plane area as is enclosed
and completely bounded by the circumference of a circle. All lines drawn in
it from one internal point to another can be deformed into one another ; any
simple closed line lying entirely within it can be deformed so as to be
evanescent, without in either case passing over the circumference ; and any
simple line from one point of the circumference to another, when regarded as
an impassable barrier, divides the surface into two portions. Such a surface
is called* simply connected.

The kind of connected surface next in point of simplicity is such a plane
area as is enclosed between and is completely bounded by the circumferences
of two concentric circles. All lines in the surface
from one point to another cannot necessarily be
deformed into one another, e.g., the lines z0az and
zj)z; a simple closed line cannot necessarily be
deformed so as to be evanescent without crossing
the boundary, e.g., the line az^bza ; and a simple
line from a point in one part of the boundary to
a point in another and different part of the
boundary, such as a line AB, does not divide the
surface into two portions but, set as an impassable barrier, it makes the
surface simply connected.

Again, on the surface of an anchor-ring, a closed line can be drawn in
two essentially distinct ways, abc, cib'c', such
that neither can be deformed so as to be evanes
cent or so as to pass continuously into the other.
If abc be made the only impassable barrier, a
line such as afty cannot be deformed so as to be
evanescent ; if ab'c' be made the only impassable
barrier, the same holds of a line such as a/3'y'.
In order to make the surface simply connected,
two impassable barriers, such as abc and ab'c',
must be set.

Surfaces, like the flat ring or the anchor-

Fig. 35.

Fig. 36.

* Sometimes the term vionadelphic is used. The German equivalent is einfach ziisammen-
hangend.

314

CROSS-CUTS AND LOOP-CUTS

[158.

ring, are called* multiply connected] the establishment of barriers has made it
possible, in each case, to modify the surface into one which is simply connected.

159. It proves to be convenient to arrange surfaces in classes according
to the character of their connection ; and these few illustrations suggest that
the classification may be made to depend, either upon the resolution of the
surface, by the establishment of barriers, into one that is simply connected,
or upon the number of what may be called independent irreducible circuits.
The former mode — that of dependence upon the establishment of barriers —
will be adopted, thus following Biemann-f- ; but whichever of the two modes
be adopted (and they are not necessarily the only modes) subsequent de
mands require that the two be brought into relation with one another.

The most effective way of securing the impassability of a barrier is to
suppose the surface actually cut along the line of the barrier. Such a section
of a surface is either a cross-cut or a loop-cut.

If the section be made through the interior of the surface from one point

Fig. 37.

of the boundary to another point of the boundary, without intersecting itself
or meeting the boundary save at its extremities, it is called a cross-cut\.
Every part of it, as it is made, is to be regarded as boundary during the
formation of the remainder ; and any cross-cut, once made, is to be regarded
as boundary during the formation of any cross-cut subsequently made.
Illustrations are given in Fig. 37.

The definition and explanation imply that the surface has a boundary.
Some surfaces, such as a complete sphere and a complete anchor-ring, do not
possess a boundary; but, as will be seen later (§§ 163, 168) from the
discussion of the evanescence of circuits, it is desirable to assign some
boundary in order to avoid merely artificial difficulties as to the numerical

* Sometimes the term polyadc.lphic is used. The German equivalent is mehrfach zusammen-
Mngcnd.

t " Grundlagen fur eine allgemeine Theorie der Functionen einer veriindeiiichen complexen
Grosse," Eiemann's Gesammelte Werke, pp. 9 — 12; "Theorie der Abel'schen Functionen," ib.,/
pp. 84—89. When reference to either of these memoirs is made, it will be by a citation "et ih^
page or pages in the volume of lliemann's Collected Works.

£ This is the equivalent used for the German word Querschnitt ; French writers use Section,
and Italian writers use Trasversale or Taglio trasversale.

159.] CONNECTION DEFINED 315

expression of the connection. This assignment usually is made by taking for
the boundary of a surface, which otherwise has no boundary, an infinitesimal
closed curve, practically a point; thus in the figure of the anchor-ring
(Fig. 36) the point a is taken as a boundary, and each of the two cross-cuts
begins and ends in a.

If the section be made through the interior of the surface from a point
not on the boundary and, without meeting the boundary or crossing itself,
return to the initial point, (so that it has the form of a simple curve lying

Fig. 38.

entirely in the surface), it is called* a loop-cut. Thus a piece can be cut
out of a bounded spherical surface by a loop-cut (Fig. 38) ; but it does
not necessarily give a separate piece when made in the surface of an
anchor-ring.

It is evident that both a cross-cut and a loop-cut furnish a double
boundary-edge to the whole aggregate of surface, whether consisting of two
pieces or of only one piece after the section.

Moreover, these sections represent the impassable barriers of the pre
liminary explanations ; and no specified form was assigned to those barriers.
It is thus possible, within certain limits, to deform a cross-cut or a loop-cut
continuously into a closely contiguous and equivalent position. If, for
instance, two barriers initially coincide over any finite length, one or other
can be slightly deformed so that finally they intersect only in a point ; the
same modification can therefore be made in the sections.

The definitions of simple connection and of multiple connection will nowf*
be as follows : —

A surface is simply connected, if it be resolved into two distinct pieces by
every cross-cut; but if there be any cross-cut, which does not resolve it into
distinct pieces, the surface is multiply connected.

160. Some fundamental propositions, relating to the connection of
surfaces, may now be derived.

* This is the equivalent used for the German word Riickkehrsclmitt ; French writers use the
word Retroscction.

t Other definitions will be required, if the classification of surfaces be made to depend on
methods other than resolution by sections.

316 RESOLUTION BY CROSS-CUTS [160.

I. Each of the two distinct pieces, into which a simply connected surface S
is resolved by a cross-cut, is itself simply connected.

If either of the pieces, made by a cross-cut ab, be not simply connected,
then some cross-cut cd must be possible which will not resolve that piece into
distinct portions.

If neither c nor d lie on ab, then the obliteration of the cut ab will restore
the original surface 8, which now is not resolved by the cut cd into distinct
pieces.

If one of the extremities of cd, say c, lie on ab, then the obliteration of the
portion cb will change the two pieces into a single piece which is the original
surface 8; and 8 now has a cross-cut acd, which does not resolve it into
distinct pieces.

If both the extremities lie on ab, then the obliteration of that part of ab
which lies between c and d will change the two pieces into one ; this is the
original surface 8, now with a cross-cut acdb, which does not resolve it into
distinct pieces.

These are all the possible cases should either of the distinct pieces of 8
not be simply connected ; each of them leads to a contradiction of the simple
connection of 8', therefore the hypothesis on which each is based is untenable,
that is, the distinct pieces of 8 in all the cases are simply connected.

COROLLARY 1. A singly connected surface is resolved by n cross-cuts into
Ti+1 distinct pieces, each simply connected; and an aggregate of m simply
connected surfaces is resolved by n cross-cuts into n -f m distinct pieces each
simply connected.

COROLLARY 2. A surface that is resolved into two distinct simply con
nected pieces by a cross-cut is simply connected before the resolution.

COROLLARY 3. // a multiply connected surface be resolved into two
different pieces by a cross-cut, both of these pieces cannot be simply connected.

We now come to a theorem* of great importance : —

II. If a resolution of a surface by m cross-cuts into n distinct simply
connected pieces be possible, and also a different resolution of the same surface by
fjb cross-cuts into v distinct simply connected pieces, then m — n = fj, — v.

Let the aggregate of the n pieces be denoted by 8 and the aggregate of
the v pieces by 2 : and consider the effect on the original surface of a united
system of in + p simultaneous cross-cuts made up of the two systems of the
m and of the /j, cross-cuts respectively. The operation of this system can be
carried out in two ways : (i) by effecting the system of /u, cross-cuts on 8 and

* The following proof of this proposition is substantially due to Neumann, p. 157. Another
proof is given by Riemann, pp. 10, 11, and is amplified by Durege, Elemente der Theorie der
Functional, pp. 183 — 190 ; and another by Lippich, see Durege, pp. 190 — 197.

160.] CONNECTIVITY 317

(ii) by effecting the system of m cross-cuts on 2 : with the same result on the
original surface.

After the explanation of § 159, we may justifiably assume that the lines
of the two systems of cross-cuts meet only in points, if at all : let 8 be the
number of points of intersection of these lines. Whenever the direction of a
cross-cut meets a boundary line, the cross-cut terminates ; and if the direction
continue beyond that boundary line, that produced part must be regarded as
a new cross-cut.

Hence the new system of /u, cross-cuts applied to S is effectively equiva
lent to (j, + & new cross-cuts. Before these cuts were made, S was composed
of n simply connected pieces ; hence, after they are applied, the new arrange
ment of the original surface is made up of n + (/j, + 8) simply connected
pieces.

Similarly, the new system of m cross-cuts applied to 2 will give an
arrangement of the original surface made up of v + (m + 8) simply connected
pieces. These two arrangements are the same : and therefore

n + fj, + 8 — v + in + 8,
so that m — n = p — v.

It thus appears that, if by any system of q cross-cuts a multiply connected
surface be resolved into a number p of pieces distinct from one another and
all simply connected, the integer q — p is independent of the particular
system of the cross-cuts and of their configuration. The integer q—p is
therefore essentially associated with the character of the multiple connection
of the surface : and its invariance for a given surface enables us to arrange
surfaces according to the value of the integer.

No classification among the multiply connected surfaces has yet been
made : they have merely been defined as surfaces in which cross-cuts can be
made that do not resolve the surface into distinct pieces.

It is natural to arrange them in classes according to the number of cross
cuts which are necessary to resolve the surface into one of simple connection
or a number of pieces each of simple connection.

For a simply connected surface, no such cross-cut is necessary: then
q = 0, p=l, and in general q — p = — l. We shall say that the connectivity*
is unity. Examples are furnished by the area of a plane circle, and by a
spherical surface with one hole^.

A surface is called doubly- connected when, by one appropriate cross-cut,
the surface is changed into a single surface of simple connection : then q = 1,
p = 1 for this particular resolution, and therefore in general, q—p = Q. We

* Sometimes order of connection, sometimes adelphic order ; the German word, that is used,
is Grundzahl.

+ The hole is made to give the surface a boundary (§ 163).

318 EFFECT OF CROSS-CUTS [160.

shall say that the connectivity is 2. Examples are furnished by a plane ring
and by a spherical surface with two holes.

A surface is called triply-connected when, by two appropriate cross-cuts,
the surface is changed into a single surface of simple connection : then q = 2,
p = l for this particular resolution and therefore, in general, q — p = l. We
shall say that the connectivity is 3. Examples are furnished by the surface
of an anchor- ring with one hole in it*, and by the surfaces -f- in Figure 39, the
surface in (2) not being in one plane but one part beneath another.

Fig. 39.

And, in general, a surface will be said to be ^V-ply connected or its
connectivity will be denoted by N, if, by N — 1 appropriate cross-cuts, it can
be changed into a single surface that is simply connected |. For this
particular resolution q = N—\, p = l: and therefore in general

q-p = N-2,
or N = q-p + 2.

Let a cross-cut I be drawn in a surface of connectivity N. There are
two cases to be considered, according as it does not or does divide the surface
into distinct pieces.

First, let the surface be only one piece after I is drawn : and let its
connectivity then be N'. If in the original surface q cross-cuts (one of
which can, after the preceding proposition, be taken to be I) be drawn
dividing the surface into p simply connected pieces, then

N = q-p+ 2.

To obtain these p simply connected pieces from the surface after the cross-cut
I, it is evidently sufficient to make the q — 1 original cross-cuts other than I ;
that is, the modified surface is such that by q — 1 cross-cuts it is resolved into
p simply connected pieces, and therefore

Hence N' = N — 1, or the connectivity of the surface is diminished by unity.

* The hole is made to give the surface a boundary (§ 163).

t Riemann, p. 89.

J A few writers estimate the connectivity of such a surface as N- 1, the same as the number
of cross-cuts which can change it into a single surface of the simplest rank of connectivity : the
estimate in the text seems preferable.

160.]

ON THE CONNECTIVITY

319

Secondly, let the surface be two pieces after I is drawn, of connectivities
Ni and N2 respectively. Let the appropriate JVj — 1 cross-cuts in the former,
and the appropriate N2 — 1 in the latter, be drawn so as to make each a
simply connected piece. Then, together, there are two simply connected
pieces.

To obtain these two pieces from the original surface, it will suffice to
make in it the cross-cut I, the Ni — I cross-cuts, and the N2—l cross-cuts,
that is, 1 + (Ni. — 1) + (N* — 1) or Nj, + N2 — 1 cross-cuts in all. Since these,
when made in the surface of connectivity N, give two pieces, we have

and therefore

If one of the pieces be simply connected, the connectivity of the other is JV;
so that, if a simply connected piece of surface be cut off a multiply connected
surface, the connectivity of the remainder is unchanged. Hence :

III. If a cross-cut be made in a surface of connectivity N and if it do
not divide it into separate pieces, the connectivity of the modified surface is
N—l; but if it divide the surface into two separate pieces of connectivities N!
and N«, then Nl + N2 = N+ 1.

Illustrations are shewn, in Fig. 40, of the effect of cross-cuts on the two
surfaces in Fig. 39.

IV. In the same way it may be proved that, if s cross-cuts be made in a
surface of connectivity N and divide it into r+l separate pieces (where r^.s)
of connectivities N1} N2, ..., Nr+l respectively, then

a more general result including both of the foregoing cases.

Thus far we have been considering only cross-cuts : it is now necessary
to consider loop-cuts, so far as they affect the connectivity of a surface in
which they are made.

320 EFFECT OF LOOP-CUTS [160.

A loop-cut is changed into a cross-cut, if from A any point of it a cross-cut
be made to any point C in a boundary-curve of the
original surface, for CAbdA (Fig. 41) is then evi- /•

dently a cross-cut of the original surface ; and CA is
a cross-cut of the surface, which is the modification
of the original surface after the loop-cut has been
made. Since, by definition, a loop-cut does not
meet the boundary, the cross-cut CA does not
divide the modified surface into distinct pieces ;
hence, according as the effect of the loop-cut is, \ Fi8- 41-

or is not, that of making distinct pieces, so will
the effect of the whole cross-cut be, or not be, that of making distinct pieces.

161. Let a loop-cut be drawn in a surface of connectivity N; as before
for a cross-cut, there are two cases for consideration, according as the loop-cut
does or does not divide the surface into distinct pieces.

First, let it divide the surface into two distinct pieces, say of connectivities
N! and N2 respectively. Change the loop-cut into a cross-cut of the original
surface by drawing a cross-cut in either of the pieces, say the second, from a
point in the course of the loop-cut to some point of the original boundary.
This cross-cut, as a section of that piece, does not divide it into distinct
pieces: and therefore the connectivity is now N? (= N2 — 1). The effect of
the whole section, which is a single cross-cut, of the original surface is to
divide it into two pieces, the connectivities of which are JVa and N2' : hence,
by S 160, III.,

and therefore N1 + Na

If the piece cut out be simply connected, say JVj. = 1, then the connectivity
of the remainder is N + 1. But such a removal of a simply connected piece
by a loop-cut is the same as making a hole in a continuous part of the
surface : and therefore the effect of making a simple hole in a continuous part
of a surface is to increase by unity the connectivity of the surface.

If the piece cut out be doubly connected, say N: = 2, then the connect
ivity of the remainder is N, the same as the connectivity of the original
surface. Such a portion would be obtained by cutting out a piece with a
hole in it which, so far as concerns the original surface, would be the same as
merely enlarging the hole — an operation that naturally would not affect
the connectivity.

Secondly, let the loop -cut not divide the surface into two distinct pieces :
and let N' be the connectivity of the modified surface. In this modified
surface make a cross-cut k from any point of the loop-cut to a point of the
boundary: this does not divide it into distinct pieces and therefore the
connectivity after this last modification is N' -I. But the surface thus

161.] ON THE CONNECTIVITY 321

finally modified is derived from the original surface by the single cross-cut,
constituted by the combination of k with the loop-cut : this single cross-cut
does not divide the surface into distinct pieces and therefore the connectivity
after the modification is N — 1. Hence

that is, JV' = N, or the connectivity of a surface is not affected by a loop-cut
which does not divide the surface into distinct pieces.

Both of these results are included in the following theorem : —

V. If after any number of loop-cuts made in a surface of connectivity
N, there be r + 1 distinct pieces of surface, of connectivities JV^ JV2, ..., Nr+lt
then

N, + N3 + ...... + JVr+1 = JV+2r.

Let the number of loop-cuts be s. Each of them can be changed into a
cross-cut of the original surface, by drawing in some one of the pieces, as may
be convenient, a cross-cut from a point of the loop-cut to a point of a
boundary ; this new cross-cut does not divide the piece in which it is drawn
into distinct pieces. If k such cross-cuts (where k may be zero) be drawn in
the piece of connectivity Nm, the connectivity becomes Nm', where

N ' — N~ — If-

•" m — •*•• m I" j
r+l r+l r+l

hence 2 Nm' = 2 Nm-2k= X Nm - s.

m=\ m-\ m=l

We now have s cross-cuts dividing the surface of connectivity JV into r + l
distinct pieces, of connectivities JV/, JV/, ..., JV/, Nr+1' ; and therefore, by
§ 160, IV.,

so that JVj + JV2 + . . . 4- Nr+1 = JV + 2r.

This result could have been obtained also by combination and repetition
of the two results obtained for a single loop-cut.

Thus a spherical surface with one hole in it is simply connected : when
n — l other different holes* are made in it, the edges of the holes being
outside one another, the connectivity of the surface is increased by n— 1,
that is, it becomes n. Hence a spherical surface with n holes in it is n-ply
connected.

162. Occasionally, it is necessary to consider the effect of a slit made in
the surface.

If the slit have neither of its extremities on a boundary (and therefore no
point on a boundary) it can be regarded as the limiting form of a loop-cut
which makes a hole in the surface. Such a slit therefore (§ 161) increases the
connectivity by unity.

* These are holes in the surface, not holes bored through the volume of the sphere ; one of
the latter would give two holes in the surface.

F- 21

BOUNDARIES [162.

If the slit have one extremity (but no other point) on a boundary, it can
be regarded as the limiting form of a cross-cut, which returns
on itself as in the figure, and cuts off a single simply con- /
nected piece. Such a slit therefore (§ 160, III.) leaves the
connectivity unaltered.

If the slit have both extremities on boundaries, it ceases \
to be merely a slit : it is a cross-cut the effect of which on Fl8- 42-

the connectivity has been obtained. We do not regard such
sections as slits.

163. In the preceding investigations relative to cross-cuts and loop-cuts,
reference has continually been made to the boundary of the surface con
sidered.

The boundary of a surface consists of a line returning to itself, or of a
system of lines each returning to itself. Each part of such a boundary-line
as it is drawn is considered a part of the boundary, and thus a boundary-line
cannot cut itself and pass beyond its earlier position, for a boundary cannot
be crossed: each boundary-line must therefore be a simple curve*.

Most surfaces have boundaries : an exception arises in the case of closed
surfaces whatever be their connectivity. It was stated (§ 159) that a
boundary is assigned to such a surface by drawing an infinitesimal simple
curve in it or, what is the same thing, by making a small hole. The
advantage of this can be seen from the simple example of a spherical
surface.

When a small hole is made in any surface the connectivity is increased
by unity : the connectivity of the spherical surface after the hole is made is
unity, and therefore the connectivity of the complete spherical surface
must be taken to be zero.

The mere fact that the connectivity is less than unity, being that of the
simplest connected surfaces with which we have to deal,
is not in itself of importance. But let us return for a
moment to the suggested method of determining the
connectivity by means of the evanescence of circuits
without crossing the boundary. When the surface is
the complete spherical surface (Fig. 43), there are two
essentially distinct ways of making a circuit C evan
escent, first, by making it collapse into the point a, Fig. 43.
secondly by making it expand over the equator and

then collapse into the point b. One of the two is superfluous : it introduces
an element of doubt as to the mode of evanescence unless that mode be
specified a specification which in itself is tantamount to an assignment of

* Also a line not returning to itself may be a boundary ; it can be regarded as the limit of a
simple curve when the area becomes infinitesimal.

163.] EFFECT OF CROSS-CUTS ON BOUNDARIES 323

boundary. And in the case of multiply connected surfaces the absence of
boundary, as above, leads to an artificial reduction of the connectivity by
unity, arising not from the greater simplicity of the surface but from the
possibility of carrying out in two ways the operation of reducing any circuit
to given circuits, which is most effective when only one way is permissible.
We shall therefore assume a boundary assigned to such closed surfaces as in
the first instance are destitute of boundary.

164. The relations between the number of boundaries and the connect
ivity of a surface are given by the following propositions.

I. The boimdary of a simply connected surface consists of a single line.

When a boundary consists of separate lines, then a cross-cut can be made
from a point of one to a point of another. By proceeding from
P, a point on one side of the cross-cut, along the boundary
ac...cVwe can by a line lying wholly in the surface reach a
point Q on the other side of the cross-cut : hence the parts of
the surface on opposite sides of the cross-cut are connected.
The surface is therefore not resolved into distinct pieces by the
cross-cut.

A simply connected surface is resolved into distinct pieces Fig. 44.
by each cross-cut made in it : such a cross-cut as the foregoing
is therefore not possible, that is, there are not separate lines which make up
its boundary. It has a boundary : the boundary therefore consists of a single
line.

II. A cross-cut either increases by unity or diminishes by unity the number
of distinct boundary -lines of a multiply connected surface.

A cross-cut is made in one of three ways : either from a point a of one
boundary-line A to a, point b of another boundary-line B ; or from a point a
of a boundary-line to another point a' of the same boundary-line ; or from a
point of a boundary-line to a point in the cut itself.

If made in the first way, a combination of one edge of the cut, the
remainder of the original boundary A, the other edge of the cut and the
remainder of the original boundary B taken in succession, form a single
piece of boundary ; this replaces the two boundary-lines A and B which
existed distinct from one another before the cross-cut was made. Hence the
number of lines is diminished by unity. An example is furnished by a plane
ring (ii., Fig. 37, p. 314).

If made in the second way, the combination of one edge of the cut with
the piece of the boundary on one side of it makes one boundary-line, and the
combination of the other edge of the cut with the other piece of the boundary
makes another boundary-line. Two boundary-lines, after the cut is made,

21—2

324 NUMBER OF BOUNDARY-LINES [164.

replace a single boundary-line, which existed before it was made : hence the
number of lines is increased by unity. Examples are furnished by the cut
surfaces in Fig. 40, p. 319.

If made in the third way, the cross-cut may be considered as constituted
by a loop-cut and a cut joining the loop-cut to the boundary. The boundary-
lines may now be considered as constituted (Fig. 41, p. 320) by the closed
curve ABD and the closed boundary abda'c'e'...eca; that is, there are now
two boundary-lines instead of the single boundary-line ce...e'c'c in the uncut
surface. Hence the number of distinct boundary-lines is increased by unity.

COROLLARY. A loop-cut increases the number of distinct boundary-lines
by two.

This result follows at once from the last discussion.

III. The number of distinct boundary-lines of a surface of connectivity N
is N — 2k, where k is a positive integer that may be zero.

Let m be the number of distinct boundary-lines ; and let N — 1 appro
priate cross-cuts be drawn, changing the surface into a simply connected
surface. Each of these cross-cuts increases by unity or diminishes by unity
the number of boundary-lines ; let these units of increase or of decrease be
denoted by e^ e2, ..., €#_!. Each of the quantities e is + 1 ; let k of them be
positive, and N — 1 — k negative. The total number of boundary-lines is

therefore

m + k-(N-l-k).

The surface now is a single simply connected surface, and there is therefore
only one boundary-line ; hence

m + k-(N-l-k) = l,
so that m = N — 2k ;

and evidently k is an integer that may be zero.

COROLLARY 1. A closed surface with a single boundary-line* is of odd
connectivity.

For example, the surface of an anchor-ring, when bounded, is of con
nectivity 3; the surface, obtained by boring two holes through the volume
of a solid sphere, is, when bounded, of connectivity 5.

If the connectivity of a closed surface with a single boundary be 2p + 1,
the surface is often said-f- to be of class p (§ 178, p. 349.)

COROLLARY 2. If the number of distinct boundary lines of a surface of
connectivity N be N, any loop-cut divides the surface into two distinct pieces.

After the loop-cut is made, the number of distinct boundary-lines is
N+2; the connectivity of the whole of the cut surface is therefore not less

* See § 159.

t The German word is Geschlecht ; French writers use the word genre, and Italians genere.

164.] LHUILIER'S THEOREM 325

than N+2. It has been proved that a loop-cut, which does not divide the
surface into distinct pieces, does not affect the connectivity ; hence as the
connectivity has been increased, the loop-cut must divide the surface into
two distinct pieces. It is easy, by the result of § 161, to see that, after the
loop-cut is made, the sum of connectivities of the two pieces is N+2, so
that the connectivity of the whole of the cut surface is equal to N + 2.

Note. Throughout these propositions, a tacit assumption has been made,
which is important for this particular proposition when the surface is the
means of representing the variable. The assumption is that the surface is
bifacial and not unifacial ; it has existed implicitly throughout all the
geometrical representations of variability : it found explicit expression in
§ 4 when the plane was brought into relation with the sphere : and a cut
in a surface has been counted a single cut, occurring in one face, though it
would have to be counted as two cuts, one on each side, were the surface
unifacial.

The propositions are not necessarily valid, when applied to unifacial
surfaces. Consider a surface made out of a long rectangular slip of paper,
which is twisted once (or any odd number of times) and then has its ends
fastened together. This surface is of double connectivity, because one
section can be made across it which does not divide it into separate pieces ;
it has only a single boundary-line, so that Prop. III. just proved does not
! apply. The surface is unifacial ; and it is possible, without meeting the
boundary, to pass continuously in the surface from a point P to another
point Q which could be reached merely by passing through the material
at P.

We therefore do not retain unifacial surfaces for consideration.

165. The following proposition, substantially due to Lhuilier*, may be
taken in illustration of the general theory.

If a closed surface of connectivity 2N + 1 (or of class N) be divided by
circuits into any number of simply connected portions, each in the form of a
curvilinear polygon, and if F be the number of polygons, E be the number of
edges and S the number of angular points, then

2N=2 + JE-F-S.

Let the edges E be arranged in systems, a system being such that any
lino in it can be reached by passage along some other line or lines of the
system ; let k be the number of such systems -f. To resolve the surface into a
number of simply connected pieces composed of the F polygons, the cross-cuts
will be made along the edges ; and therefore, unless a boundary be assigned

* Gergonne, Ann. de Math., t. iii, (1813), pp. 181—186; see also Mobius, Ges. Werke, t. ii,
p. 468. A circuit is defined in § 166.

t The value of k is 1 for the proposition and is greater than 1 for the Corollary.

326 LHUILIER'S THEOREM [165.

to the surface in each system of lines, the first cut for any system will be a
loop-cut. We therefore take k points, one in each system as a boundary ;
the first will be taken as the natural boundary of the surface, and the
remaining k—\, being the limiting forms of k — 1 infinitesimal loop-cuts,
increase the connectivity of the surface by k — 1, that is, the connectivity now
is 2N+k.

The result of the cross-cuts is to leave F simply connected pieces : hence
Q, the number of cross-cuts, is given by

At every angular point on the uncut surface, three or more polygons are
contiguous. Let Sm be the number of angular points, where m polygons are
contiguous; then

Again, the number. of edges meeting at each of the S3 points is three, atl
each of the $4 points is four, at each of the $5 points is five, and so on ; hence,
in taking the sum 3$3 + 4$4 + 5$5 + . . ., each edge has been counted twice, once
for each extremity. Therefore

Consider the composition of the extremities of the cross-cuts ; the number
of the extremities is 2Q, twice the number of cross-cuts.

Each of the k points furnishes two extremities; for each such point
is a boundary on which the initial cross-cut for each of the systems must
begin and must end. These points therefore furnish 2k extremities.

The remaining extremities occur in connection with the angular points.
In making a cut, the direction passes from a boundary along an edge, past
the point along another edge and so on, until a boundary is reached ; so that
on the first occasion when a cross-cut passes through a point, it is made along
two of the edges meeting at the point. Every other cross-cut passing through
that point must begin or end there, so that each of the S3 points will furnish
one extremity (corresponding to the remaining one cross-cut through the
point), each of the $4 points will furnish two extremities (corresponding to
the remaining two cross-cuts through the point), and so on. The total
number of extremities thus provided is

S3 + 2St+3S5 + ...
Hence 2Q = 2k + 83 + 2St + 3S6+ ...

or Q = k + E-S,

which combined with Q = 2N + k + F - 2,
leads to the relation 2N=2 + E-F-S.

165.] CIRCUITS ON CONNECTED SURFACES 327

The simplest case is that of a sphere, when Euler's relation F + S =• E + 2
is obtained. The case next in simplicity is that of an anchor-ring, for which
the relation is F+ S = E.

COROLLARY. If the result of making the cross-cuts along the various edges
be to give the F polygons, not simply connected areas but areas of connectivities
jYj + 1, jV2 + l, ..., Np+1 respectively, then the connectivity of the original
surface is given by

166. The method of determining the connectivity of a surface by means
of a system of cross-cuts, which resolve it into one or more simply connected
pieces, will now be brought into relation with the other method, suggested
in § 159, of determining the connectivity by means of irreducible circuits.

A closed line drawn on the surface is called a circuit.

A circuit, which can be reduced to a point by' continuous deformation
without crossing the boundary, is called reducible ; a circuit, which cannot be
so reduced, is called irreducible.

An irreducible circuit is either (i) simple, when it cannot without crossing
the boundary be deformed continuously into repetitions of one or more
circuits ; or (ii) multiple, when it can without crossing the boundary be
deformed continuously into repetitions of a single circuit ; or (iii) compound,
when it can without crossing the boundary be deformed continuously into
combinations of different circuits, that may be simple or multiple. The
distinction between simple circuits and compound circuits, that involve no
multiple circuits in their combination, depends upon conventions adopted for
each particular case.

A circuit is said to be reconcileable with the system of circuits into a
combination of which it can be continuously deformed.

If a system of circuits be reconcileable with a reducible circuit, the
system is said to be reducible.

As there are two directions, one positive and the other negative, in which
a circuit can be described, and as there are possibilities of repetitions and of
compositions of circuits, it is clear that circuits can be represented by linear
algebraical expressions involving real quantities and having merely numerical
coefficients.

Thus a reducible circuit can be denoted by 0.

If a simple irreducible circuit, positively described, be denoted by a, the
same circuit, negatively described, can be denoted by — a.

The multiple circuit, which is composed of m positive repetitions of the
simple irreducible circuit a, would be denoted by ma ; but if the m repetitions
were negative, the multiple circuit would be denoted by — ma.

328 CIRCUITS [106.

A compound circuit, reconcileable with a system of simple irreducible
circuits a1} a2, ..., an would be denoted by m1a1 + m2a2-\- ... + mnan, where
mj, m2, ..., mn are positive or negative integers, being the net number of
positive or negative descriptions of the respective simple irreducible circuits.

The condition of the reducibility of a system of circuits al, «2, ..., an,
each one of which is simple and irreducible, is that integers m1} m.2, ..., mn
should exist such that

m^j + m2a2 + . . . + mnan = 0,

the sign of equality in this equation, as in other equations, implying that
continuous deformation without crossing the boundary can change into one
another the circuits, denoted by the symbols on either side of the sign.

The representation of any compound circuit in terms of a system of
independent irreducible circuits is unique : if there were two different
expressions, they could be equated in the foregoing sense and this would
imply the existence of a 'relation

P& + p.2a2 + . . . +pnan = 0,
which is excluded by the fact that the system is irreducible.

Further, equations can be combined linearly, provided that the coefficients
of the combinations be merely numerical.

167. In order, then, to be in a position to estimate circuits on a multiply
connected surface, it is necessary that an irreducible system of irreducible
simple circuits should be known, such a system being considered complete
when every other circuit on the surface is reconcileable with the system.

Such a system is not necessarily unique ; and it must be proved that, if
more than one complete system be obtainable, any circuit can be reconciled with
each system.

First, the number of simple irreducible circuits in any complete system
must be tlie same for the same surface.

Let a1} ..., ap; and b1} ..., bn; be two complete systems. Because a1} ...,
ap constitute a complete system, every circuit of the system of circuits b is
reconcileable with it ; that is, integers ra# exist, such that

br = mlral + m.2ra.2 + . . . + mprap,

for r = 1, 2, ..., n. If n were >p, then by combining linearly each equation
after the first p equations with those p equations, and eliminating al, ..., ap
from the set of p + 1 equations, we could derive n —p relations of the form

M^ + M,b2 + . . . + Mnbn = 0,

where the coefficients M, being determinants the constituents of which are
integers, would be integers. The system of circuits b is irreducible, and there
are therefore no such relations ; hence n is not greater than p.

167.] ON CONNECTED SURFACES 329

Similarly, by considering the reconciliation of each circuit a with the
irreducible system of circuits b, it follows that p is not greater than n.

Hence p and n are equal to one another. And, because each system is a
complete system, there are integers A and B such that
ar = Arlbi + Ar2b.2 4- • • • + Arnbn (r = I, ...,
bs = Bg^ + Bs.2a2 + . . . -I- BmOn (s = l, ...,

The determinant of the integers A is equal to + 1 ; likewise the deter
minant of the integers B.

Secondly, let x be a circuit reconcileable with the system of circuits a : it is
reconcileable with any other complete system of circuits.

Since x is reconcileable with the system a, integers m1} ..., mn can be

found such that

x = ??i1«1 + . . . + mnan.

Any other complete system of n circuits b is such that the circuits a can
be expressed in the form

ar = Anbj. + ... + Arnbn , (r = 1, . . ., n),
where the coefficients A are integers ; and therefore

n n n

x = b1'2 mrArl 4- 62 S mrArz + . . . + bn X mrArn

r=l r=l r=l

= gri&i + gr2&a + ~'+qnl>n,

where the coefficients q are integers, that is, x is reconcileable with the
complete system of circuits b.

168. It thus appears that for the construction of any circuit on a surface,
it is sufficient to know some one complete system of simple irreducible
circuits. A complete system is supposed to contain the smallest possible
number of simple circuits : any one which is reconcileable with the rest is
omitted, so that the circuits of a system may be considered as independent.
Such a system is indicated by the following theorems : —

I. No irreducible simple circuit can be drawn on a simply connected
surface*.

If possible, let an irreducible circuit G be drawn in a simply connected
surface with a boundary B. Make a loop-cut along C, and change it into a
cross-cut by making a cross-cut A from some point of C to a point of B ;
this cross-cut divides the surface into two simply connected pieces, one of
which is bounded by B, the two edges of A, and one edge of the cut along C,
and the other of which is bounded entirely by the cut along C.

The latter surface is smaller than the original surface ; it is simply

connected and has a single boundary. If an irreducible simple circuit can

be drawn on it, we proceed as before, and again obtain a still smaller simply

connected surface. In this way, we ultimately obtain an infinitesimal

* All surfaces considered are supposed to be bounded.

330 RELATIONS BETWEEN CONNECTIVITY [168.

element ; for every cut divides the surface, in which it is made, into
distinct pieces. Irreducible circuits cannot be drawn in this element ; and
therefore its boundary is reducible. This boundary is a circuit in a larger
portion of the surface : the circuit is reducible so that, in that larger portion
no irreducible circuit is possible and therefore its boundary is reducible.
This boundary is a circuit in a still larger portion, and the circuit is
reducible : so that in this still larger portion no irreducible circuit is possible
and once more the boundary is reducible.

Proceeding in this way, we find that no irreducible simple circuit is
possible in the original surface.

COROLLARY. No irreducible circuit can be drawn on a simply connected
surface.

II. A complete system of irreducible simple circuits for a surface of
connectivity N contains N— I simple circuits, so that every other circuit on the
surface is reconcileable with that system.

Let the surface be resolved by cross-cuts into a single simply connected
surface: N— 1 cross-cuts will be necessary. Let CD be
any one of them : and let a and b be two points on the /e

opposite edges of the cross-cut. Then since the surface is L n

simply connected, a line can be drawn in the surface from
a to b without passing out of the surface or without
meeting a part of the boundary, that is, without meeting
any other cross-cut. The cross-cut CD ends either in Fis- 45-

another cross-cut or in a boundary; the line ae...fb
surrounds that other cross-cut or that boundary as the case may be : hence,
if the cut CD be obliterated, the line ae...fba is irreducible on the surface in
which the other N — 2 cross-cuts are made. But it meets none of those cross
cuts; hence, when they are all obliterated so as to restore the unresolved
surface of connectivity N, it is an irreducible circuit. It is evidently riot
a repeated circuit; hence it is an irreducible simple circuit. Hence the
line of an irreducible simple circuit on an unresolved surface is given by
a line passing from a point on one edge of a cross-cut in the resolved
surface to a point on the opposite edge.

Since there are N -I cross-cuts, it follows that N —1 irreducible simple
circuits can thus be obtained: one being derived in the foregoing manner
from each of the cross-cuts, which are necessary to render the surface simply
connected. It is easy to see that each of the irreducible circuits on an
unresolved surface is, by the cross-cuts, rendered impossible as a circuit on
the resolved surface.

But every other irreducible circuit C is reconcileable with the N—l
circuits, thus obtained. If there be one not reconcileable with these N-l
circuits, then, when all the cross-cuts are made, the circuit C is not rendered

168.]

AND IRREDUCIBLE CIRCUITS

331

impossible, if it be not reconcileable with those which are rendered impossible
by the cross-cuts : that is, there is on the resolved surface an irreducible
circuit. But the resolved surface is simply connected, and therefore no
irreducible circuit can be drawn on it : hence the hypothesis as to C, which
leads to this result, is not tenable.

Thus every other circuit is reconcileable with the system of N — 1 circuits :
and therefore the system is complete*.

This method of derivation of the circuits at once indicates how far a
system is arbitrary. Each system of cross-cuts leads to a complete system of
irreducible simple circuits, and vice versa ; as the one system is not unique,
so the other system is not unique.

For the general question, Jordan's memoir, Des contours traces sur les surfaces,
Liouville, 2me Ser., t. xi., (1866), pp. 110—130, may be consulted.

Ex. 1. On a doubly connected surface, one irreducible simple circuit can be drawn.
It is easily obtained by first resolving the surface into one that is simply connected —
a single cross-cut CD is effective for this purpose — and then by drawing a curve aeb in the

Fig. 46, (i).

surface from one edge of the cross-cut to the other. All other irreducible circuits on the
unresolved surface are reconcileable with the circuit aeba.

Ex. 2. On a triply- connected surface, two independent irreducible circuits can be

Fig. 46, (ii).

* If the number of independent irreducible simple circuits be adopted as a basis for the
definition of the connectivity of a surface, the result of the proposition would be taken as the
definition : and the resolution of the surface into one, which is simply connected, would then be
obtained by developing the preceding theory in the reverse order.

332

DEFORMATION

[168.

drawn. Thus in the figure Cl and C2 will form a complete system. The circuits C3 and (74
are also irreducible : they can evidently be deformed into C^ and <72 and reducible circuits
by continuous deformation : in the algebraical notation adopted, we have

C3=C1 + C2, Ci=Cl-C.2.

Ex. 3. Another example of a triply connected surface is given in Fig. 47. Two irredu
cible simple circuits are Cv and C%. Another irreducible circuit is C3; this can be

Fig. 47.

reconciled with Cl and C.2 by drawing the point a into coincidence with the intersection
of Cj and (72, and the point c into coincidence with the same point.

Ex. 4. As a last example, consider the surface of a solid sphere with n holes bored
through it. The connectivity is 2n + 1 : hence 2n independent irreducible simple circuits

Fig. 48.

can be drawn on the surface. The simplest complete system is obtained by taking 2n
curves : made up of a set of n, each round one hole, and another set of n, each through
one hole.

A resolution of this surface is given by taking cross-cuts, one round each hole (making
the circuits through the holes no longer possible) and one through each hole (making the
circuits round the holes no longer possible).

The simplest case is that for which n= 1 : the surface is equivalent to the anchor-ring.

169. Surfaces are at present being considered in view of their use as a
means of representing the value of a complex variable. The foregoing inves
tigations imply that surfaces can be classed according to their connectivity ;
and thus, having regard to their designed use, the question arises as to
whether all surfaces of the same connectivity arc equivalent to one another,
so as to be transformable into one another.

169.] OF CONNECTED SURFACES 333

Moreover, a surface can be physically deformed and still remain suitable for
representation of the variable, provided certain conditions are satisfied. We
thus consider geometrical transformation as well as physical deformation ; but
we are dealing only with the general results and not with the mathematical
relations of stretching and bending, which are discussed in treatises on
Analytical Geometry*.

It is evident that continuity is necessary for both : discontinuity would
imply discontinuity in the representation of the variable. Points that are
contiguous (that is, separated only by small distances measured in the surface)
must remain contiguous -f*: and one point in the unchanged surface must
correspond to only one point in the changed surface. Hence in the continuous
deformation of a surface there may be stretching and there may be bending ;
but there must be no tearing and there must be no joining.

For instance, a single untwisted ribbon, if cut, comes to be simply connected. If a twist
through 180° be then given to one end and that end be then joined to the other, we shall
have a once- twisted ribbon, which is a surface with only one face and only one edge;
it cannot be looked upon as an equivalent of the former surface.

A spherical surface with a single hole can have the hole stretched and the surface
flattened, so as to be the same as a bounded portion of a plane : the two surfaces are
equivalent to one another. Again, in the spherical surface, let a large indentation be
made : let both the outer and the inner surfaces be made spherical ; and let the mouth of
the indentation be contracted into the form of a long, narrow hole along a part of a great
circle. When each point of the inner surface is geometrically moved so that it occupies the
position of its reflexion in the diametral plane of the hole, the final form§ of the whole
surface is that of a two-sheeted surface with a junction along a line : it is a spherical
winding-surface, and is equivalent to the simply connected spherical surface.

170. It is sufficient, for the purpose of representation, that the two
surfaces should have a point-to-point transformation : it is not necessary
that physical deformation, without tears or joins, should be actually possible.
Thus a ribbon with an even number of twists would be as effective as a
limited portion of a cylinder, or (what is the same thing) an untwisted ribbon :
but it is not possible to deform the one into the other physically |.

It is easy to see that either deformation or transformation of the kind
considered will change a bifacial surface into a bifacial surface ; that it will
not alter the connectivity, for it will not change irreducible circuits into

* See, for instance, Frost's Solid Geometry, (3rd ed.), pp. 342 — 352.

t Distances between points must be measured along the surface, not through space ; the
distance between two points is a length which one point would traverse before reaching the
position of the other, the motion of the point being restricted to take place in the surface.
Examples will arise later, in Biemann's surfaces, in which points that are contiguous in space
are separated by finite distances on the surface.

§ Clifford, Coll. Hath. Papers, p. 250.

J The difference between the two cases is that, in physical deformation, the surfaces are the
surfaces of continuous matter and are impenetrable ; while, in geometrical transformation, the
surfaces may be regarded as penetrable without interference with the continuity.

334 DEFORMATION OF SURFACES [170.

reducible circuits, and the number of independent irreducible circuits
determines the connectivity: and that it will not alter the number of boundary
curves, for a boundary will be changed into a boundary. These are necessary
relations between the two forms of the surface : it is not difficult to see that
they are sufficient for correspondence. For if, on each of two bifacial surfaces
with the same number of boundaries and of the same connectivity, a complete
system of simple irreducible circuits be drawn, then, when the members of the
systems are made to correspond in pairs, the full transformation can be effected
by continuous deformation of those corresponding irreducible circuits. It
therefore follows that : —

The necessary and sufficient conditions, that two bifacial surfaces may be
equivalent to one another for the representation of a variable, are that tlie two
surfaces should be of the same connectivity and should have the same number of
boundaries.

As already indicated, this equivalence is a geometrical equivalence :
deformation may be (but is not of necessity) physically possible.

Similarly, the presence of one or of several knots in a surface makes no
essential difference in the use of the surface for representing a variable. Thus
a long cylindrical surface is changed into an anchor-ring when its ends are
joined together ; but the changed surface would be equally effective for
purposes of representation if a knot were tied in the cylindrical surface before
the ends are joined.

But it need hardly be pointed out that though surfaces, thus twisted or
knotted, are equivalent for the purpose indicated, they are not equivalent for
all topological enumerations.

Seeing that bifacial surfaces, with the same connectivity and the same
number of boundaries, are equivalent to one another, it is natural to adopt, as
the surface of reference, some simple surface with those characteristics; thus
for a surface of connectivity 2p + 1 with a single boundary, the surface of a
solid sphere, bounded by a point and pierced through with p holes, could be
adopted.

Klein calls* such a surface of reference a Normal Surface.

It has been seen that a bounded spherical surface and a bounded simply connected
part of a plane are equivalent — they are, moreover, physically deformable into one
another.

An untwisted closed ribbon is equivalent to a bounded piece of a plane with one hole
in it — they are deformable into one another : but if the ribbon, previous to being closed,
have undergone an even number of twists each through 180°, they are still equivalent
but are not physically deformable into one another. Each of the bifacial surfaces is
doubly connected (for a single cross-cut renders each simply connected) and each of them

* Ueber Riemann's Theorie der algebraischen Functionen und ihrer Integrate, (Leipzig,
Teubner, 1882), p. 26.

170.] REFERENCES 335

has two boundaries. If however the ribbon, previous to being closed, have imdcrgone
an odd number of twists each through 180°, the surface thus obtained is not equivalent to
the single-holed portion of the plane ; it is unifacial arid has only one boundary.

A spherical surface pierced in n-\-l holes is equivalent to a bounded portion of the
plane with n holes ; each is of connectivity n + 1 and has n + 1 boundaries. The spherical
surface can be deformed into the plane surface by stretching one of its holes into the form
of the outside boundary of the plane surface.

Ex. Prove that the surface of a bounded anchor-ring can be physically deformed into
the surface in Fig. 47, p. 332.

For continuation and fuller development of the subjects of the present chapter, the
following references, in addition to those which have been given, will be found useful :

Klein, Math. Ann., t. vii, (1874), pp. 548—557; ib., t. ix, (1876), pp. 476—482.

Lippich, Math. Ann., i. vii, (1874), pp. 212 — 229 ; Wiener Sitzungsb., t. Ixix, (ii),
(1874), pp. 91—99.

Durege, Wiener Sitzungsb., t. Ixix, (ii), (1874), pp. 115—120; and section 9 of his
treatise, quoted on p. 316, note.

Neumann, chapter vii of his treatise, quoted on p. 5, note.

Dyck, Math. Ann., t. xxxii, (1888), pp. 457—512, ib., t. xxxvii, (1890), pp. 273—316;

at the beginning of the first part of this investigation, a valuable series of references
is given.

Dingeldey, Topologische Studien, (Leipzig, Teubner, 1890).

CHAPTER XV.

RIEMANN'S SURFACES.

171. THE method of representing a variable by assigning to it a position
in a plane or on a sphere is effective when properties of uniform functions of
that variable are discussed. But when multiform functions, or integrals of
uniform functions occur, the method is effective only when certain parts of
the plane are excluded, due account being subsequently taken of the effect of
such exclusions; and this process, the extension of Cauchy's method, was
adopted in Chapter IX.

There is another method, referred to in § 100 as due to Riemann, of an
entirely different character. In Riemann's representation, the region, in
which the variable z exists, no longer consists of a single plane but of a
number of planes ; they are distinct from one another in geometrical concep
tion, yet, in order to preserve a representation in which the value of the
variable is obvious on inspection, the planes are infinitesimally close to one
another. The number of planes, often called sheets, is the same as the
number of distinct values (or branches) of the function w for a general
argument z and, unless otherwise stated, will be assumed finite; each sheet
is associated with one branch of the function, and changes from one branch
of the function to another are effected by making the ^-variable change
from one sheet to another, so that, to secure the possibility of change
of sheet, it is necessary to have means of passage from one sheet to another.
The aggregate of all the sheets is a surface, often called a Riemanns
Surface.

For example, consider the function

w=z* + (z-I}~*,

the cube roots being independent of one another. It is evidently a nine-valued function ;
the number of sheets in the appropriate Eiemann's surface is therefore nine.

The branch-points are 2 = 0, z = l, 2=00. Let o> and a denote a cube-root of unity,
independently of one another ; then the values of z* can be represented in the form

171.]

EXAMPLES OF RIEMANN's SURFACES

337

ill -A - 4

23, C023", co22*; and the values of (2-!) 3 can be represented in the form (2-!) ,

^•(z - \ ) ~ 3} 0 (« - 1) » The nine values of w can be symbolically expressed as follows : —

Fig. 49.

Fig. 50.

where the symbols opposite to w give the coefficients of z3 and of (2- 1) 3 respectively.

Now when 2 describes a small simple circuit positively round the origin, the groups
in cyclical order are u\, w2, w3; w4, w5, w6; wr, w8, io9. And therefore, in the immediate
vicinity of the origin, there must be means of passage to enable
the 2-point to make the corresponding changes from sheet to —
sheet. Taking a section of the whole surface near the origin ~
so as to indicate the passages and regarding the right-hand
sides as the part from which the 2-variable moves when it —
describes a circuit positively, the passages must be in character as
indicated in Fig. 49. And it is evident that the further descrip
tion of small simple circuits round the origin will, with these passages, lead to the proper
values : thus %, which after the single description is the value of w4, becomes w6 after
another description and it is evident that a point in the w-0 sheet passes into the w6 sheet.

When 2 describes a small simple circuit positively round the point 1, the groups in cyclical
order are wlt ^4, %; w2, w5, ws; w3, w6, w9: and therefore,
in the immediate vicinity of the point 1, there must be ~
means of passage to render possible the corresponding changes
of 2 from sheet to sheet. Taking a section as before near the ~
point 1 and with similar convention as to the positive direc
tion of the 2-path, the passages must be in character as
indicated in Fig. 50.

Similarly for infinitely large values of 2.

If then the sheets can be so joined as to give these possibilities of passage and also
give combinations of them corresponding to combinations of the simple paths indicated,
then there will be a surface to any point of which will correspond one and only one value
of w : and when the value of w is given for a point 2 in an ordinary plane of variation,
then that value of w will determine the sheet of the surface in which the point 2 is to
be taken. A surface will then have been constructed such that the function w, which is
multiform for the single-plane representation of the variable, is uniform for variations
in the many-sheeted surface.

Again, for the simple example arising from the two-valued function, defined by
the equation

w = {(z-a}(z-b}(z-c}}-\

the branch-points are a, b, c, oo ; and a small simple circuit round any one of these
four points interchanges the two values. The Riemann's surface is two-sheeted and
there must be means of passage between the two sheets in the vicinity of a, that of b,
that of c and at the infinite part of the plane.

These examples are sufficient to indicate the main problem. It is the
construction of a surface in which the independent variable can move so
F. 22

338 SHEETS OF HIEMANN'S SURFACE [171.

that, for variations of z in that surface, the multiformity of the function is
changed to uniformity. From the nature of the case, the character of the
surface will depend on the character of the function : and thus, though all the
functions are uniform within their appropriate surfaces, these surfaces are
widely various. Evidently for uniform functions of z the appropriate surface
on the above method is the single plane already adopted.

172. The simplest classes of functions for which a Riemaim's surface is
useful are (i) those called (§ 94) algebraic functions, that is, multiform functions
of the independent variable denned by an algebraical equation of the form

which is of finite degree, say n, in w, and (ii) those usually called Abelian
functions, which arise through integrals connected with algebraic functions.

Of such an algebraic function there are, in general, n distinct values ; but
for the special values of z, that are the branch-points, two or more of the
values coincide. The appropriate Riemann's surface is composed of n sheets ;
one branch, and only one branch, of w is associated with a sheet. The
variable z, in its relation to the function, is determined not merely by its
modulus and argument but also by its sheet ; that is, in the language of the
earlier method, we take account of the path by which z acquires a value. The
particular sheet in which z lies determines the particular branch of the
function. Variations of #, which occur within a sheet and do not coincide
with points lying in regions of passage between the sheets, lead to variations
in the value of the branch of w associated with the sheet ; a return to an
initial value of z, by a path that nowhere lies within a region of passage,
leaves the ^-point in the same sheet as at first and so leads to the initial
branch (and to the initial value of the branch) of w. But a return to an
initial value of z by a path, which, in the former method of representation,
would enclose a branch-point, implies a change of the branch of the function
according to the definite order prescribed by the branch-point. Hence the
final value of the variable z on the Riemann's surface must lie in a sheet that
is different from that of the initial (and algebraically equal) value ; and
therefore the sheets must be so connected that, in the immediate vicinity of
branch-points, there are means of passage from one sheet to another, securing
the proper interchanges of the branches of the function as defined by the
equation.

173. The first necessity is therefore the consideration of the mode in
which the sheets of a Riemann's surface are joined : the mode is indicated by
the theorem that sheets of a Riemann's surface are joined along lines.

The junction might be made either at a point, as with two spheres in
contact, or by a common portion of a surface, as with one prism lying on

173.] JOINED ALONG BRANCH-LINES 339

another, or along lines ; but whatever the character of the junction be, it
must be such that a single passage across it (thereby implying entrance to
the junction and exit from it) must change the sheet of the variable.

If the junction were at a point, then the £- variable could change from one
sheet into another sheet, only if its path passed through that point : any
other closed path would leave the z- variable in its original sheet. A small
closed curve, infinites! rn ally near the point and enclosing it and no other
branch-point, is one which ought to transfer the variable to another sheet
because it encloses a branch-point : and this is impossible with a point-junction
when the path does not pass through the point. Hence a junction at a point
only is insufficient to provide the proper means of passage from sheet to
sheet.

If the junction were effected by a common portion
of surface, then a passage through it (implying an
entrance into that portion and an exit from it) ought to
change the sheet. But, in such a case, closed contours .-'--''
can be constructed which make such a passage without Fi8- 51>

enclosing the branch-point a : thus the junction would cause a change of
sheet for certain circuits the description of which ought to leave the
z- variable in the original sheet. Hence a junction by a continuous area of
surface does not provide proper means of passage from sheet to sheet.

The only possible junction which remains is a line.

The objection in the last case does not apply to a closed • / '^

contour which does not contain the branch-point ; for the /.--"''

line cuts the curve twice and there are therefore two Fig. 52.

crossings ; the second of them makes the variable return to the sheet which
the first crossing compelled it to leave.

Hence the junction between any two sheets takes place along a line.

Such a line is called* a branch-line. The branch -points of a multiform
function lie on the branch-lines, after the foregoing explanations ; and a
branch-line can be crossed by the variable only if the variable change its
sheet at crossing, in the sequence prescribed by the branch-point of the
function which lies on the line. Also, the sequence is reversed when the
branch-line is crossed in the reversed direction.

Thus, if two sheets of a surface be connected along a branch-line, a point which
crosses the line from the first sheet must pass into the second and a point which crosses
the line from the second sheet must pass into the first.

Again, if, along a common direction of branch-line, the first sheet of a surface
be connected with the second, the second with the third, and the third with

* Sometimes cross-line, sometimes branch-section. The German title is Verzweigungschnitt;
the French is lignc de passage ; see also the note on the equivalents of branch-point, p. 15.

22—2

340 PROPERTIES OF BRANCH-LINES [173.

the first, a point which crosses the line from the first sheet in one direction must pass
into the second sheet, but if it cross the line in the other direction it must pass into
the third sheet.

A branch -point does not necessarily affect all the branches of a function :
when it affects only some of them, the corresponding property of the Riemann's
surface is in evidence as follows. Let z=a determine a branch-point affecting,
say, only r branches. Take n points a, one in each of the sheets ; and through
them draw n lines cab, having the same geometrical position in the respective
sheets. Then in the vicinity of the point a in each of the n sheets, associated
with the r affected branches, there must be means of passage from each one
to all the rest of them ; and the lines cab can conceivably be the branch-lines
with a properly established sequence. The point a does not affect the other
n — r branches : there is therefore no necessity for means of passage in the
vicinity of a among the remaining n — r sheets. In each of these remaining
sheets, the point a and the line cab belong to their respective sheets alone :
for them, the point a is not a branch-point and the line cab is not a branch-
line.

174. Several essential properties of the branch-lines are immediate
inferences from these conditions.

I. A free end of a branch-line in a surface is a branch-point.

Let a simple circuit be drawn round the free end so small as to enclose no
branch-point (except the free end, if it be a branch-point). The circuit meets
the branch-line once, and the sheet is changed because the branch-line is
crossed ; hence the circuit includes a branch-point which therefore can be
only the free end of the line.

Note. A branch-line may terminate in the boundary of the surface,
and then the extremity need not be a branch-point.

II. When a branch-line extends beyond a branch-point lying in its course,
the sequence of interchange is not the same on the two sides of the point.

If the sequence of interchange be the same on the two sides of the branch
point, a small circuit round the point would first cross one part of the branch-
line and therefore involve a change of sheet and then, in its course, would
cross the other part of the branch-line in the other direction which, on the
supposition of unaltered sequence, would cause a return to the initial sheet.
In that case, a circuit round the branch-point would fail to secure the proper
change of sheet. Hence the sequence of interchange caused by the branch-
line cannot be the same on the two sides of the point.

III. If two branch-lines with different sequences of interchange have a
common extremity, that point is either a branch-point or an extremity of at
least one other branch-line.

174.] SYSTEM OF BRANCH-LINES 341

If the point be not a branch-point, then a simple curve enclosing it, taken
so small as to include no branch-point, must
leave the variable in its initial sheet. Let A
be such a point, AB and AC be two branch-
lines having A for a common extremity ; let ., A ,.• — ^ «

the sequence be as in the figure, taken for a F.

simple case ; and suppose that the variable

initially is in the rth sheet. A passage across AB makes the variable
pass into the sth sheet. If there be no branch-line between AB and AC
having an extremity at A, and if neither n nor m be s, then the passage
across AC makes no change in the sheet of the variable and, therefore, in
order to restore r before AB, at least one branch-line must lie in the angle
between AC and AB, estimated in the positive trigonometrical sense.

If either n or m, say n, be s, then after passage across AC, the point is in
the mt\i sheet ; then, since the sequences are not the same, m is not r and
there must be some branch-line between AC and AB to make the point
return to the rth sheet on the completion of the circuit.

If then the point A be not a branch-point, there must be at least one
other branch-line having its extremity at A. This proves the proposition.

COROLLARY 1. If both of two branch-lines extend beyond a point of inter
section, which is not a branch-point, no sheet of the surface has both of them for
branch-lines.

COROLLARY 2. If a change of sequence occur at any point of a branch-
line, then either that point is a branch-point or it lies also on some other
branch-line.

COROLLARY 3. No part of a branch-line with only one branch-point on it
can be a closed curve.

It is evidently superfluous to have a branch-line without any branch-point
on it.

175. On the basis of these properties, we can obtain a system of branch-
lines satisfying the requisite conditions which are : —

(i) the proper sequences of change from sheet to sheet must be
secured by a description of a simple circuit round a branch
point : if this be satisfied for each of the branch-points, it
will evidently be satisfied for any combination of simple circuits,
that is, for any path whatever enclosing one or more branch
points.

(ii) the sheet, in which the variable re-assumes its initial value after
describing a circuit that encloses no branch-point, must be the
initial sheet.

342 SYSTEM OF BRANCH-LINES [175.

In the ^-plane of Cauchy's method, let lines be drawn from any point I, not
a branch-point in the first instance, to each of the branch-points, as in fig. 19,
p. 156, so that the joining lines do not meet except at /: and suppose the
w-sheeted Riemann's surface to have branch-lines coinciding geometrically
with these lines, as in § 173, and having the sequence of interchange for
passage across each the same as the order in the cycle of functional values
for a small circuit round the branch-point at its free end. No line (or part
of a line) can be a closed curve ; the lines need not be straight, but they
will be supposed drawn as direct as possible to the points in angular
succession.

The first of the above requisite conditions is satisfied by the establish
ment of the sequence of interchange.

To consider the second of the conditions, it is convenient to divide
circuits into two kinds, (a) those which exclude /, (/3) those which include /,
no one of either kind (for our present purpose) including a branch-point.

A closed circuit, excluding I and all the branch-points, must intersect a
branch-line an even number of times, if
it intersect the line in real points. Let
the figure (fig. 54) represent such a case :
then the crossings at A and B counter
act one another and so the part be
tween A and B may without effect be
transferred across IB3 so as not to cut
the branch-line at all. Similarly for
the points C and D : and a similar
transference of the part now between
C and D may be made across the
branch-line without effect: that is, the
circuit can, without effect, be changed
so as not to cut the branch-line IBS at all. A similar change can be made
for each of the branch-lines : and so the circuit can, without effect, be changed
into one which meets no branch-line and therefore, on its completion, leaves
the sheet unchanged.

A closed circuit, including / but no branch-point, must meet each branch-
line an odd number of times. A change similar in character to that in
the previous case may be made for each branch-line : and without affecting
the result, the circuit can be changed so that it meets each branch-line only
once. Now the effect produced by a branch-line on the function is the same
as the description of a simple loop round the branch-point which with /
determines the branch-line : and therefore the effect of the circuit at present
contemplated is, after the transformation which does not affect the result, the
same as that of a circuit, in the previously adopted mode of representation,

175.] FOR A SURFACE

enclosing all the branch-points. But, by Cor. III. of § 90, the effect of a
circuit which encloses all the branch-points (including z = GO , if it be a
branch-point) is to restore the value of the function which it had at the
beginning of the circuit : and therefore in the present case the effect is to
make the point return to the sheet in which it lay initially.

It follows therefore that, for both kinds of a closed circuit containing no
branch-point, the effect is to make the ^-variable return to its initial sheet
on resuming its initial value at the close of the circuit.

Next, let the point / be a branch-point ; and let it be joined by lines,
as direct* as possible, to each of the other branch -points in angular succes
sion. These lines will be regarded as the branch-lines ; and the sequence of
interchange for passage across any one is made that of the interchange pre
scribed by the branch-point at its free extremity.

The proper sequence of change is secured for a description of a simple
closed circuit round each of the branch-points other than /. Let a small
circuit be described round /; it meets each of the branch-lines once and
therefore its effect is the same as, in the language of the earlier method of
representing variation of z, that of a circuit enclosing all the branch-points
except 7. Such a circuit, when taken on the Neumann's sphere, as in Cor.
III., § 90 and Ex. 2, § 104, may be regarded in two ways, according as one or
other of the portions, into which it divides the area of the sphere, is regarded
as the included area; in one way, it is a circuit enclosing all the branch
points except /, in the other it is a circuit enclosing / alone and no other
branch-point. Without making any modification in the final value of w, it
can (by § 90) be deformed, either into a succession of loops round all the
branch-points save one, or into a loop round that one ; the effect of these two
deformations is therefore the same. Hence the effect of the small closed
circuit round / meeting all the branch-lines is the same as, in the other mode
of representation, that of a small curve round / enclosing no other branch
point ; and therefore the adopted set of branch- lines secures the proper
sequence of change of value for description of a circuit round /.

The first of the two necessary conditions is therefore satisfied by the
present arrangement of branch-lines.

The proof, that the second of the two necessary conditions is also satisfied
by the present arrangement of branch-lines, is similar to that in the preceding
case, save that only the first kind of circuit of the earlier proof is possible.

Jt thus appears that a system of branch-lines can be obtained which
secures the proper changes of sheet for a multiform function : and therefore
Riemann's surfaces can be constructed for such a function, the essential
property being that over its appropriate surface an otherwise multiform
function of the variable is a uniform function.

* The reason for this will appear in §§ 183, 184.

344 EXAMPLES [175.

The multipartite character of the function has its influence preserved by
the character of the surface to which the function is referred : the surface,
consisting of a number of sheets joined to one another, may be a multiply
connected surface.

In thus proving the general existence of appropriate surfaces, there has
remained a large arbitrary element in their actual construction : moreover,
in particular cases, there are methods of obtaining varied configurations of
branch-lines. Thus the assignment of the n branches to the n sheets has
been left unspecified, and is therefore so far arbitrary : the point I, if not a
branch-point, is arbitrarily chosen and so there is a certain arbitrariness of
position in the branch -lines. Naturally, what is desired is the simplest
appropriate surface : the particularisation of the preceding arbitrary qualities
is used to derive a canonical form of the surface.

176. The discussion of one or two simple cases will help to illustrate the
mode of junction between the sheets, made by branch-lines.

The simplest case of all is that in which the surface has only a single
sheet: it does not require discussion.

The case next in simplicity is that in which the surface is two-sheeted :
the function is therefore two- valued and is consequently defined by a
quadratic equation of the form

Lua + 2Mu + N = 0,

where L and M are uniform functions of z. When a new variable w is
introduced, defined by Lu + M=w, so that values of iv and of u correspond
uniquely, the equation is

It is evident that every branch-point of u is a branch-point of w, and
vice versa ; hence the Riemann's surface is the same for the two equations.
Now any root of P (z) of odd degree is a branch-point of iv. If then

where R (z} is a product of only simple factors, every factor of R (z) leads to
a branch-point. If the degree of R (z} be even, the number of branch-points
for finite values of the variable is even and z = oo is not a branch-point ; if the
degree of R(z) be odd, the number of branch -points for finite values of the
variable is odd and z = oo is a branch-point : in either case, the number of
branch-points is even.

There are only two values of w, and the Riemann's surface is two-sheeted:
crossing a branch-line therefore merely causes a change of sheet. The free
ends of branch-lines are branch-points ; a small circuit round any branch
point causes an interchange of the branches w, and a circuit round any two
branch-points restores the initial value of w at the end and therefore leaves
the variable in the same sheet as at the beginning. These are the essential
requirements in the present case ; all of them are satisfied by taking each

176.] OF RIEMANN'S SURFACES 345

branch-line as a line connecting two (and only two) of the branch-points. The
ends of all the branch -lines are free : and their number, in this method, is
one-half that of the (even) number of branch-points. A small circuit round
a branch-point meets a branch-line once and causes a change of sheet ; a
circuit round two (and not more than two) branch -points causes either no
crossing of branch-line or an even number of crossings and therefore restores
the variable to the initial sheet.

A branch-line is, in this case, usually drawn in the form of a straight line
when the surface is plane : but this form is not essential and all that is
desirable is to prevent intersections of the branch-lines.

Note. Junction between the sheets along a branch-line is easily secured.
The two sheets to be joined are cut along the branch-line. One edge of the
cut in the upper sheet, say its right edge looking along the section, is joined
to the left edge of the cut in the lower sheet ; and the left edge in the upper
sheet is joined to the right edge in the lower.

A few simple examples will illustrate these remarks as to the sheets : illustrations of
closed circuits will arise later, in the consideration of integrals of multiform functions.

Ex. 1. Let w* = A(z-a)(z-b},

so that a and b are the only branch-points. The surface is two-sheeted : the line ab may
be made the branch-line. In Fig. 55 only part of the upper sheet is shewn*, as likewise
only part of the lower sheet. Continuous lines imply what is visible ; arid dotted lines
what is invisible, on the supposition that the sheets are opaque.

The circuit, closed in the surface and passing round 0, is made up of OJK in the upper
sheet : the point crosses the branch-line and then passes into the lower sheet, where it
describes the dotted line KLH : it then meets and crosses the branch-line at If, passes
into the upper sheet and in that sheet returns to 0. Similarly of the line ABC, the part
AB lies in the lower sheet, the part EC in the upper : of the line DG the part DE lies in
the upper sheet, the part EFG in the lower, the piece FG of this part being there visible
beyond the boundary of the retained portion of the upper surface.

Ex. 2. Let Aw?2 = z3-a3.

The branch-points (Fig. 56) are A ( = a), B ( = ««), (7( = aa2), where a is a primitive cube
root of unity, and 2 = 00. The branch -lines can be made by BC, Ace ; and the two-
sheeted surface will be a surface over which w is uniform. Only a part of each sheet
is shewn in the figure; a section also is made at M across the surface, cutting the branch -
line A QO .

Ex. 3. Let wm=zn,

where n and TO are prime to each other. The branch-points are z = 0 and 2=00 ; and the
branch-line extends from 0 to QO . There are m sheets ; if we associate them in order with
the branches ws, where

wa=re

for s=l, 2, ..., TO, then the first sheet is connected with the second forwards, the second
with the third forwards, and so on ; the mth being connected with the first forwards.

* The form of the three figures in the plate opposite p. 346 is suggested by Holzmiiller, Ein-
fiihrung in die Theorie der isogonalen Vericandschaften und der confomien AbbUdimgen, (Leipzig,
Teubner, 1882), in which several illustrations are given.

346

SPHERICAL RIEMANN'S SURFACE

[176.

The surface is sometimes also called a winding-surface; and a branch-point such as
z—0 on the surface, where a number m of sheets pass into one another in succession, is
also called a winding-point of order m— 1 (see p. 15, note). An illustration of the surface
for m = 3 is given in Fig. 57, the branch-line being cut so as to shew the branching : what
is visible is indicated by continuous lines ; what is in the second sheet, but is invisible, is
indicated by the thickly dotted line ; what is in the third sheet, but is invisible, is indic
ated by the thinly dotted line.

Ex. 4. Consider a three-sheeted surface having four branch-points at a, b, c, d ; and
let each point interchange two branches, say, w.2, w3 at a ; iv^ w3ai b ; w2, w3 at c ; wlt w2

at d ; the points being as in Fig. 58. It is easy to verify that these branch-points
satisfy the condition that a circuit, enclosing them all, restores the initial value of w.

The branching of the sheets may be made as in the figure, the integers on the two sides
of the line indicating the sheets that are to be joined along the line.

A canonical form for such a surface can be derived from the more general case given
later (in §§ 186—189).

Ex. 5. Shew that, if the equation

be of degree n in w and be irreducible, all the n sheets of the surface are connected, that
is, it is possible by an appropriate path to pass from any sheet to any other sheet.

177. It is not necessary to limit the surface representing the variable to
a set of planes; and, indeed, as with uniform functions, there is a convenience
in using the sphere for the purpose.

We take n spheres, each of diameter unity, touching the Riemann's plane
surface at a point A ; each sphere is regarded as the stereographic projection
of a plane sheet, with regard to the other extremity A' of the spherical
diameter through A. Then, the sequence of these spherical sheets being
the same as the sequence of the plane sheets, branch-points in the plane
surface project into branch-points on the spherical surface : branch -lines be
tween the plane sheets project into branch-lines between the spherical sheets
and are terminated by corresponding points ; and if a branch-line extend in
the plane surface to z=co, the corresponding branch-line in the spherical
surface is terminated at A'.

A surface will thus be obtained consisting of n spherical sheets; like
the plane Riemann's surface, it is one over which the n-valued function is a
uniform function of the position of the variable point.

Fig.

M — =-00

To face p. 346.

Fig. 57.

177.] CONNECTIVITY OF A RIEMANN's SURFACE 347

But also the connectivity of the n-sheeted spherical surface is the same as
that of the n-sheeted plane surface with which it is associated.

In fact, the plane surface can be mechanically changed into the spherical
surface without tearing, or repairing, or any change except bending and
compression: all that needs to be done is that the n plane sheets shall be
bent, without making any change in their sequence, each into a spherical
form, and that the boundaries at infinity (if any) in the plane-sheet shall
be compressed into an infinitesimal point, being the South pole of the cor
responding spherical sheet or sheets. Any junctions between the plane
sheets extending to infinity are junctions terminated at the South pole. As
the plane surface has a boundary, which, if at infinity on one of the sheets, is
therefore not a branch-line for that sheet, so the spherical surface has a
boundary which, if at the South pole, cannot be the extremity of a branch-
line.

178. We proceed to obtain the connectivity of a Riemann's surface : it
is determined by the following theorem : —

Let the total number of branch-points in a Riemann's n-sheeted surface be
r ; and let the number of, branches of the function interchanging at the first
point be ml, the number interchanging at the second be m.2, and so on. Then
the connectivity of the surface is

fl-2n + 3,
where fl denotes m,1 + m2 + ... + mr — r.

Take* the surface in the bounded spherical form, the connectivity N of
which is the same as that of the plane surface : and let the boundary be a
small hole A in the outer sheet. By means of cross-cuts and loop-cuts, the
surface can be resolved into a number of distinct simply connected pieces.

First, make a slice bodily through the sphere, the edge in the
outside sheet meeting A and the direction of the
slice through A being chosen so that none of the
branch -points lie in any of the pieces cut off. Then n
parts, one from each sheet and each simply connected,
are taken away. The remainder of the surface has a
cup-like form ; let the connectivity of this remainder
be M.

This slice has implied a number of cuts.

The cut made in the outside sheet is a cross-cut,
because it begins and ends in the boundary A. It
divides the surface into two distinct pieces, one being
the portion of the outside sheet cut off, and this piece is simply connected ;

* The proof is founded on Neumann's, pp. 108 — 172.

348 CONNECTIVITY OF A SURFACE [178.

hence, by Prop. III. of § 160, the remainder has its connectivity still repre
sented by N.

The cuts in all the other sheets, caused by the slice, are all loop-cuts,
because they do not anywhere meet the boundary. There are n — 1 loop-
cuts, and each cuts off a simply connected piece ; and the remaining surface
is of connectivity M. Hence, by Prop. V. of § 161,

M + n - 1 = N + 2 (n - 1),
and therefore M = N+n—l.

In this remainder, of connectivity M, make r — 1 cuts, each of which
begins in the rim and returns to the rim, and is to be made through the n
sheets together ; and choose the directions of these cuts so that each of the
r resulting portions of the surface contains one (and only one) of the branch
points.

Consider the portion of the surface which contains the branch-point
where ml sheets of the surface are connected. The ml connected sheets
constitute a piece of a winding-surface round the winding-point of order
ml — 1 ; the remaining sheets are unaffected by the winding-point, and
therefore the parts of them are n — m^ distinct simply connected pieces.
The piece of winding- surface is simply connected ; because a circuit, that
does not contain the winding-point, is reducible without passing over the
winding-point, and a circuit, that does contain the winding-point, is reducible
to the winding-point, so that no irreducible circuit can be drawn. Hence
the portion of the surface under consideration consists of n — ml + 1 distinct
simply connected pieces.

Similarly for the other portions. Hence the total number of distinct
simply connected pieces is

r

2 (n - mq + 1)

9 = 1

r

= m — 2 mq + r
l-i

= nr — fl.

But in the portion of connectivity M each of the r — 1 cuts causes, in
each of the sheets, a cut passing from the boundary and returning to the
boundary, that is, a cross-cut. Hence there are n cross-cuts from each of the
r—\ cuts, and therefore n (r— 1) cross-cuts altogether, made in the portion of
surface of connectivity M.

The effect of these n(r — 1) cross-cuts is to resolve the portion of con
nectivity M into nr — £l distinct simply connected pieces ; hence, by § 160,

M = n (r - 1) - (nr - H) + 2,

and therefore N = M — (n — 1) = n - 2n + 3,

the connectivity of the Riemann's surface.

178.] CLASS OF A SURFACE 349

r

The quantity H, having the value 2 (mq — 1), may be called the rami-

</=i

fication of the surface, as indicating the aggregate sum of the orders of
the different branch-points.

Note. The surface just considered is a closed surface to which a point
has been assigned for boundary; hence, by Cor. I., Prop. III., § 164, its
connectivity is an odd integer. Let it be denoted by 2p + 1 ; then

2p = ft - 2/i + 2,

and 2p is the number of cross-cuts which change the Riemann's surface into
one that is simply connected.

The integer p is often called (Cor. I., Prop. III., § 164) the class of the
Riemann's surface; and the equation

f(w, z) = 0

is said to be of class p, when p is the class of the associated Riemann's
surface.

Ex. 1. When the equation is

w> = \(z-a}(z-b\

we have a two-sheeted surface, ?t = 2. There are two branch- points, z = a and z = b; but
2=00 is not a branch-point ; so that r=2. At each of the branch-points the two values are
interchanged, so that m1 = 2, ??i2 = 2; thus Q = 2. Hence the connectivity =2-4 + 3 = 1,
that is, the surface is simply connected.

The surface can be deformed, as in the example in § 169, into a sphere.
Ex. 2. When the equation is

we have ?t = 2. There are four branch-points, viz., et, e2, e3, oc , so that r = 4 ; and at each
of them the two values of w are interchanged, so that mg = 2 (for 5 = 1,2, 3, 4), and therefore
Q = 8- 4 = 4. Hence the connectivity is 4- 4 + 3, that is, 3 ; and the value of p is unity.

Similarly, the surface associated with the equation

where U(z] is a rational, integral, algebraical function of degree 2«i - 1 or of degree 2»i,
is of connectivity 2wi + l ; so that p = m. The equation

W2==(1_22)(1_^2)

is of class p=\. The case next in importance is that of the algebraical equation leading to
the hyperelliptic functions, when (/"is either a quintic or a sextic ; and then p = 2.

Ex. 3. Obtain the connectivity of the Riemann's surface associated with the equation

w3 + ^ — 3awz = 1 ,
where a is a constant, (i) when a is zero, (ii) when a is different from zero.

•350 RESOLUTION OF A RIEMANN's SURFACE [178.

Ex. 4. Shew that, if the surface associated with the equation

f(w,z) = 0,

have p. boundary-lines instead of one, and if the equation have the same branch-points
as in the foregoing proposition, the connectivity is Q-

179. The consideration of irreducible circuits on the surface at once
reveals the multiple connection of the surface, the numerical measure of
which has been obtained. In a Riemann's surface, a simple
closed circuit cannot be deformed over a branch-point. Let
A be a branch-point, and let AE... be the branch-line
having a free end at A. Take a curve ...CED... crossing
the branch-line at E and passing into a sheet different
from that which contains the portion CE ; and, if possible,
let a slight deformation of the curve be made so as to transfer the portion
CE across the branch-point A. In the deformed position, the curve
...C'E'D' '... does not meet the branch-line; there is, consequently, no
change of sheet in its course near A and therefore E'D'..., which is the
continuation of ...C'E', cannot be regarded as the deformed position of ED.
The two paths are essentially distinct ; and thus the original path cannot be
deformed over the branch-point.

It therefore follows that continuous deformation of a circuit over a
branch-point on a Riemann's surface is a geometrical impossibility.

Ex. Trace the variation of the curve CED, as the point E moves up to A and then
returns along the other side of the branch-line.

Hence a circuit containing two or more of the branch-points is irreducible ;
but a circuit containing all the branch-points is equivalent to a circuit that
contains none of them, and it is therefore reducible.

If a circuit contain only one branch-point, it can be continuously deformed
so as to coincide with the point on each sheet and therefore, being deformable
into a point, it is a reducible circuit. An illustration has already occurred in
the case of a portion of winding-surface containing a single winding-point
(p. 348); all circuits drawn on it are reducible.

It follows from the preceding results that the Riemann's surface associated
with a multiform function is generally one of multiple connection ; we shall
find it convenient to know how it can be resolved, by means of cross-cuts, into
a simply connected surface. The representative surface will be supposed a
closed surface with a single boundary ; its connectivity, necessarily odd, being
2/) + l, the number of cross-cuts necessary to resolve the surface into one
that is simply connected is 2p ; when these cuts have been made, the simply
connected surface then obtained will have its boundary composed of a single
closed curve.

179.]

BY CROSS-CUTS

351

One or two simple examples of resolution of special Riemann's surfaces will be useful
in leading up to the general explanation ; in the examples it will be shewn how, in
conformity with § 168, the resolving cross-cuts render irreducible circuits impossible.

Ex. 1. Let *he equation be

w1 = A(z-d)(z-b'](z-c}(z-d\

where a, b, c, d are four distinct points, all of finite modulus. The surface is two-sheeted ;
each of the points a, b, c, d is a branch-point where the two values of w interchange ; and
so the surface, assumed to have a single boundary, is triply connected, the value of p
being unity. The branch-lines are two, each connecting a pair of branch-points ; let them
be ab and cd.

Two cross-cuts are necessary and sufficient to resolve the surface into one that is
simply connected. We first make a cross-cut,
beginning at the boundary S, (say it is in the
upper sheet), continuing in that sheet and re
turning to J3, so that its course encloses the
branch-line ab (but not cd) and meets no branch-
line. It is a cross-cut, and not a loop-cut, for it
begins and ends in the boundary ; it is evidently
a cut in the upper sheet alone, and does not
divide the surface into distinct portions ; and,
once made, it is to be regarded as boundary for
the partially cut surface.

The surface in its present condition is con
nected : and therefore it is possible to pass from one edge to the other of the cut just
made. Let P be a point on it ; a curve that passes from one edge to the other is indicated
by the line PQR in the upper sheet, RS in the lower, and SP in the upper. Along this
line make a cut, beginning at P and returning to P ; it is a cross-cut, partly in the
upper sheet and partly in the lower, and it does not divide the surface into distinct
portions.

Two cross-cuts in the triply connected surface have now been made ; neither of them,
as made, divides the surface into distinct portions, and each of them when made reduces
the connectivity by one unit ; hence the surface is now simply connected. It is easy to
see that the boundary consists of a single line not intersecting itself; for beginning
at P, we have the outer edge of PUT, then the inner edge of 2'QltSP, then the inner
edge of PTB, and then the outer edge of PSRQP, returning to P.

The required resolution has been effected.

Before the surface was resolved, a number of irreducible circuits could be drawn ; a
complete system of irreducible circuits is composed of two, by § 168. Such a system may
be taken in various ways ; let it be composed of a simple curve C lying in the upper sheet
and containing the points a and b, and a simple curve D, lying partly in the upper
and partly in the lower sheet and containing the points a and c ; each of these curves
is irreducible, because it encloses two branch-points. Every other irreducible circuit
is reconcileable with these two ; the actual reconciliation in particular cases is effected
most simply when the surface is taken in a spherical form.

The irreducible circuit C on the unresolved surface is impossible on the resolved
surface owing to the cross-cut SPQRS ; and the irreducible circuit D on the unresolved
surface is impossible on the resolved surface owing to the cross-cut PTB. It is easy
to verify that no irreducible circuit can be drawn on the resolved surface.

352

RESOLUTION

[179.

In practice, it is conveniently effective to select a complete system of irreducible
simple circuits and then to make the cross-cuts so that each of them renders one circuit
of the system impossible on the resolved surface.

Ex. 2. If the equation be

= 4:(z-e1)(z-e.t)(z-e3),

the branch-points are els e2, e3 and oo . When the two-sheeted surface is spherical, and the
branch-lines are taken to be (i) a line joining elf e.2', and (ii) a line joining e3 to the South
pole, the discussion of the surface is similar in detail to that in the preceding example.

Ex. 3. Let the equation be

t*«-4* (!-«)(*-*) <X-*)&*-«),

and for simplicity suppose that AC, X, /* are real quantities subject to the inequalities

The associated surface is two-sheeted and has a boundary assigned to it ; assuming
that its sheets are planes, we shall take some point in the finite part of the upper sheet,
not being a branch-point, as the boundary. There are six . branch-points, viz., 0, 1, K,
X, /x, co at each of which the two values of w interchange ; and so the connectivity of the
surface is 5 and its class, p, is 2. The branch-lines can be taken as three, this being
the simplest arrangement ; let them be the lines joining 0, 1 ; K, X ; /*, oo .

Four cross-cuts are necessary to resolve the surface into one that is simply connected
and has a single boundary. They may be obtained as follows.

Fig. 62.

Beginning at the boundary L, let a cut LHA be made entirely in the upper sheet
along a line which, when complete, encloses the points 0 and 1 but no other branch-points ;
let the cut return to L. This is a cross-cut and it does not divide the surface into
distinct pieces ; hence, after it is made, the connectivity of the modified surface is 4, and
there are two boundary lines, being the two edges of the cut LHA.

Beginning at a point A in LHA, make a cut along ABC in the upper sheet until
it meets the branch-line /zoo, then in the lower sheet along CSD until it meets the
branch-line 01, and then in the upper sheet from D returning to the initial point A.
This is a cross-cut and it does not divide the surface into distinct pieces ; hence, after it
is made, the connectivity of the modified surface is 3, and it is easy to see that there
is only one boundary edge, similar to the single boundary in Ex. 1 when the surface
in that example has been completely resolved.

Make a loop-cut EFG along a line, enclosing the points K and X but no other branch
points ; and change it into a cross-cut by making a cut from E to some point B of the
boundary. This cross-cut can be regarded as BEFGE, ending at a point in its own
earlier course. As it does not divide the surface into distinct pieces, the connectivity is
reduced to 2 ; and there are two boundary lines.

179.] BY CROSS-CUTS 353

Beginning at a point G make another cross-cut GQPRG, as in the figure, enclosing
the two branch-points X and p, and lying partly in the upper sheet and partly in the lower.
It does not divide the surface into distinct pieces : the connectivity is reduced to unity
and there is a single boundary line.

Four cross-cuts have been made ; and the surface has been resolved into one that is
simply connected.

It is easy to verify :

(i) that neither in the upper sheet, nor in the lower sheet, nor partly in the
upper sheet and partly in the lower, can an irreducible circuit be drawn in the resolved
surface ; and

• (ii) that, owing to the cross-cuts, the simplest irreducible circuits in the unresolved
surface — viz. those which enclose 0, 1 ; 1, K ; *, X ; X, /i ; respectively — are rendered
impossible in the resolved surface.

The equation in the present example, and the Riemann's surface associated with it,
lead to the theory of hyperelliptic functions*.

180. The last example suggests a method of resolving any two-sheeted
surface into a surface that is simply connected.

The number of its branch-points is necessarily even, say 2p + 2. The
branch-lines can be made to join these points in pairs, so that there will be
p + l of them. To determine the connectivity (§ 178), we have n = 2 and,
since two values are interchanged at every branch-point, H = 2p -f 2 ; so
that the connectivity is 2p + 1. Then 2p cross-cuts are necessary for the
required resolution of the surface.

We make cuts round p of the branch-lines, that is, round all of them but
one ; each cut is made to enclose two branch-points, and each lies entirely in
the upper sheet. These are cuts corresponding to the cuts LHA and EFG
in fig. 62 ; and, as there, the cut round the first branch-line begins and ends
in the boundary, so that it is a cross-cut. All the remaining cuts are loop-
cuts at present. The system of p cuts we denote by a1} a2, ..., ap.

We make other p cuts, one passing from the inner edge of each of the p
cuts a already made to the branch-line which it surrounds, then in the lower
sheet to the (j) + l)th branch-line, and then in the upper sheet returning to
the point of the outer edge of the cut a at which it began. This system of
cuts corresponds to the cuts ADSGBA and GQPRG in fig. 62. Each of them
can be taken so as to meet no one of the cuts a except the one in which it
begins and ends ; and they can be taken so as not to meet one another.
This system of p cuts we denote by bl} b.2, ..., bp, where br is the cut which
begins and ends in ar. All these cuts are cross-cuts, because they begin and
end in boundary-lines.

Lastly, we make other p — 1 cuts from ar to 6.r_1} for r = 2, 3, . .., p, all in

* One of the most direct discussions of the theory from this point of view is given by Prym,
Neue Theorie der ultraelliptischen Functionen, (Berlin, Mayer and Miiller, 2nd ed., 1885).

F. 23

354 GENERAL RESOLUTION OF SURFACE [180.

the upper sheet ; no one of them, except at its initial and its final points,
meets any of the cuts already made. This system of p - 1 cuts we denote
by c% , GS, . . . , Cp .

Because br^ is a cross-cut, the cross-cut cr changes ar (hitherto a loop-
cut) into a cross-cut when cr and ar are combined into a single cut.

It is evident that no one of these cuts divides the surface into distinct
pieces; and thus we have a system of 2p cross-cuts resolving the two-sheeted
surface of connectivity 2p+I into a surface that is simply connected. The
cross-cuts in order* are

Oj, &j, C2 and aa, 62, c3 and as, bs, ...,cp and ap, bp.

181. This resolution of a general two-sheeted surface suggests f Rie-

mann's general resolution of a surface with any (finite) number of sheets.

As before, we assume that the surface is closed and has a single boundary

and that its class is p, so that 2p cross-cuts are necessary for its resolution

into one that is simply connected.

Make a cut in the surface such as not to divide it into distinct pieces;
and let it begin and end in the boundary. It is a cross-cut, say ^ ; it
changes the number of boundary-lines to 2 and it reduces the connectivity
of the cut surface to 2p.

Since the surface is connected, we can pass in the surface along a
continuous line from one edge of the cut ^ to the opposite edge. Along
this line make a cut 6j : it is a cross-cut, because it begins and ends in
the boundary. It passes from one edge of c^ to the other, that is, from one
boundary-line to another. Hence, as in Prop. II. of § 164, it does not divide '
the surface into distinct pieces; it changes the number of boundaries to 1
and it reduces the connectivity to 1p — 1.

The problem is now the same as at first, except that now only
2« — 2 cross-cuts are necessary for the required resolution. We make a
loop-cut a.2, not resolving the surface into distinct pieces, and a cross-cut
d from a point of a2 to a point on the boundary at 6j ; then Cj and a,2> taken
together, constitute a cross-cut that does not resolve the surface into distinct
pieces. It therefore reduces the connectivity to 2p — 2 and leaves two pieces
of boundary.

The surface being connected, we can pass in the surface along a continuous
line from one edge of a» to the opposite edge. Along this line we make a cut
b.2, evidently a cross-cut, passing, like h in the earlier case, from one
boundary-line to the other. Hence it does not divide the surface into

* See Neumann, pp. 178 — 182; Prym, Zur Thcorie der Fwwtionen in einer zweiblattrigen
Flfahe, (1866).

+ Riemann, Ges. Werke, pp. 122, 123 ; Neumann, pp. 182—185.

181.] BY CROSS-CUTS 355

distinct pieces; it changes the number of boundaries to 1 and it reduces
the connectivity to 2p — 3.

Proceeding in p stages, each of two cross-cuts, we ultimately obtain a
simply connected surface with a single boundary ; and the general effect on
the original unresolved surface is to have a system of cross-cuts somewhat of
the form

Fig. 63.

The foregoing resolution is called the canonical resolution of a Riemann's
surface.

Ex. 1. Construct the Riemann's surface for the equation

w3 + z3 — 3awz— 1,

both for a = 0 and for a different from zero; and resolve it by cross-cuts into a simply
connected surface with a single boundary, shewing a complete system of irreducible simple
circuits on the unresolved surface.

Ex. 2. Shew that the Riemann's surface for the equation

_
(z-c)(z-d)

is of class p = 2- indicate the possible systems of branch-lines, and, for each system,
resolve the surface by cross-cuts into a simply connected surface with a single boundary.

(Burnside.)

182. Among algebraical equations with their associated Riemann's
surfaces, two general cases of great importance and comparative simplicity
distinguish themselves.. The first is that in which the surface is two-
sheeted ; round each branch-point the two branches interchange. The
second is that in which, while the surface has a finite number of sheets
greater than two, all the branch-points are of the first order, that is, are
such that round each of them only two branches of the function interchange.
The former has already been considered, in so for as concerns the surface ;
we now proceed to the consideration of the latter.

The equation is f(w, z) = 0,

of degree n in w; and, for our present purpose, it is convenient to regard

0 as an equation corresponding to a generalised plane curve of degree n
so that no term in / is of dimensions higher than n.

The total number of branch-points has been proved, in § 98, to be

w(w-l)-28-2«,

23—2

356 DEFICIENCY [182.

where S is the number of points which are the generalisation of double
points on the curve with non-coincident tangents and K is the number
of double points on the curve with coincident tangents. Round each of
these branch-points, two branches of w interchange and only two, so that
all the numbers mq of § 178 are equal to 2 ; hence the ramification
H is

2 [n (n - 1) - 2S - 2/e} - [n (n - 1) - 2S - 2*},

that is, n=w(n-l)-28-2«.

The connectivity of the surface is therefore

w (n - 1) - 28 - 2* - 2n + 3 ;
and therefore the class p of the surface is

£(n-l)(»-2)-8-«.

Now this integer is known* as the deficiency of the curve; and therefore it
appears that the deficiency of the curve is the same as the class of the Riemann
surface associated with its equation, and also is the same as the class of its
equation.

Moreover, the number of branch-points of the original equation is fl, that

is,

n - 2

Note. The equality of these numbers, representing the deficiency and
the class, is one among many reasons that lead to the close association of
algebraic functions (and of functions dependent on them) with the theory of
plane algebraic curves, in the investigations of Nb'ther, Brill, Clebsch and
others, referred to in §§ 191, 242.

183. With a view to the construction of a canonical form of Riemann's
surface of class p for the equation under consideration, it is necessary to
consider in some detail the relations between the branches of the functions
as they are affected by the branch- points.

The effect produced on any value of the function by the description of a
small circuit, enclosing one branch-point (and only one), is known. But
when the small circuit is part of a loop, the effect on the value of the
function with which the loop begins to be described depends upon the form
of the loop; and various results (e.g. Ex. 1, § 104) are obtained by taking
different loops. In the first form (§ 175) in which the branch-lines were
established as junctions between sheets, what was done was the equivalent

* Salmon's Higher Plane Curves, §§ 44, 83; Clebsch's Vorlesungen iiber Geometrie, (edited
by Lindemann), t. i, pp. 351 — 429, the German word used instead of deficiency being Geschlecht.
The name 'deficiency' was introduced by Cayley in 1865: see Proc. Land. Math. Soc., vol. i.,
" On the transformation of plane curves."

183.] LOOPS 357

of drawing a number of straight loops, which had one extremity common to
all and the other free, and of assigning the law of junction according to the
law of interchange determined by the description of the loop. As, however,
there is no necessary limitation to the forms of branch-lines, we may draw
them in other forms, always, of course, having branch-points at their free
extremities ; and according to the variation in the form of the branch-line,
(that is, according to the variation in the form of the corresponding loop
or, in other words, according to the deformation of the loop over other
branch-points from some form of reference), there will be variation in the law
of junction along the branch-lines.

There is thus a large amount of arbitrary character in the forms of the
branch-lines, and consequently in the laws of junction along the branch-lines,
of the sheets of a Riemann's surface. Moreover, the assignment of the n
branches of the function to the n sheets is arbitrary. Hence a consider
able amount of arbitrary variation in the configuration of a Riemann's
surface is possible within the limits imposed by the invariance of its
connectivity. The canonical form will be established by making these
arbitrary elements definite.

184. After the preceding explanation and always under the hypothesis
that the branch-points are simple, we shall revert temporarily to the use of
loops and shall ultimately combine them into branch-lines.

When, with an ordinary point as origin, we construct a loop round a
branch-point, two and only two of the values of the function are affected
by that particular loop ; they are interchanged by it ; but a different form of
loop, from the same origin round the same branch-point, might affect some
other pair of values of the function.

To indicate the law of interchange, a symbol will be convenient. If the
two values interchanged by a given loop be Wi and wm, the loop will be
denoted by im ; and i and ra will be called the numbers of the symbol of that
loop.

For the initial configuration of the loops, we shall (as in § 175) take an
ordinary point 0 : we shall make loops beginning at 0, forming them in the
sequence of angular succession of the branch-points round 0 and drawing the
double linear part of the loop as direct as possible from 0 to its branch-point :
and, in this configuration, we shall take the law of interchange by a loop to
be the law of interchange by the branch-point in the loop.

In any other configuration, the symbol of a loop round any branch-point
depends upon its form, that is, depends upon the deformation over other
branch-points which the loop has suffered in passing from its initial form.
The effect of such deformation must first be obtained : it is determined by
the following lemma : —

358

MODIFICATION

[184.

When one loop is deformed over another, the symbol of the deformed loop is
unaltered, if neither of its numbers or if both of its numbers occur in the
symbol of the u